This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
2. Let E(8) be the compact set of all convergent sums ~ , 8 - ~ , = 0 or 1.
6.6. The scalar product of a compactly supported pseudo-measure S a n d a function g in B(G) is defined using the same function a : ag belongs to A(G) and ( S , g ) = ( S , a g ) ; this definition is independent of the particular a chosen.
7.2. THEOREM VI: The compact set E(0) is a set of uniqueness ifand only if 8 is a Pisot-Chabauty number. The proof of the necessity of the condition is very similar to the proof given for the real case. We can construct as in Section 2.2 a probability measure ,u carried by E(8), whose Fourier transform is
100
C:
*
6.7. Homomorphisms of groups KY) Let GI and G, be two 1.c.a. groups and h : GI + G, a continuous homomorphism. For each f in A(G,), f 0 h = g belongs to B(Gl) and IlgllBcc,) < I l f llacc,, ([151, 4.2.1, P. 79). Hence for each compactly supported pseudomeasure S on G I , we can define a pseudo-measure h(S) on G, by the rule 0 be a positive real number. The dilation with ratio O is defined on R by x + Ox. Iff is any complex-valued function on R, Tof will be the complex-valued function g defined by g(x) = f (Ox), rather than the expected f (0-'x). The product of two dilations is clearly another dilation and if V is a set of complex-valued functions on the line, we shall study the set S of all 8 > 0 such that To(%)c V . THEOREM 111: Let V be a vector space of almost-periodic functions on R. Assume that (a) V is translation-invariant; (b) for each positive E there is a relatively dense subset M ( E )of R such that each s E M ( E )is an E-almostperiod for all f in V ; (c) there is a function f in V which is not a constant. Let (To)oES be the semigroup of all dilations such that To(%)c V . Then there exists a real algebraic numberjeld K, of degree n over Q say, such that S is contained in (1) n 2, where is the set of all Pisot or Salem numbers of degree n lying in K. Before proving this theorem, a remark should be made. It may happen that the only 8 for which To(% ' ?) c V is O = 1. All the other such O are Pisot or Salem numbers. Proof: Let C be the closure of V with respect to the uniform norm. Then has the properties (a) and (b) and, using Theorem 11of Chapter IV, %?is the space of all almost-periodic functions f on the line whose spectra lie in an harmonious set A of real numbers. If O E S and O # 1 , we apply the inclusion To(@)c C to each element exp 2niAx, ilE A, of 8;we get 8A c A. Since there is a function f in V which is not a constant, there is a 2 in A which is not zero. Hence Sil c A is harmonious and so is S. We apply Theorem VI of Chapter I1 (Section 8) to S.
110
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
4. Classes of almost-periodic functions and a priori estimates on the size
r
We shall see that each harmonious subset A of the dual group of G is a coherent set of frequencies; the converse, however, is not true. PROPOSITION 1 : Let V be a class of almost periodic functions, M(E) a relatively dense set of s-almost periods for all f in V, and K(E) a compact subset of G such that K(E) + M(E) = G. For each f in V, sup If ( 2 (1 - E ) K.
Ilf ll m .
This means that, within a relative error not exceeding E, the functions in V are 'known' on the whole group as soon as they are known on K,, just as the 2n-periodic functions on the line are fully determined by their values on [O, 2n]. The proof of Proposition 1 is very simple. Each y in G can be written y = s + x, where s E M(s), x E Ke. Since each s E M(s) is a n &-almost period for each f in V, we get I f(y) - f(x)l < film; hence (f(y)l < sup (f( + f 11 a r d taking the supremum over y E G we get ProposiK.
tion 1. Let G = R and let (tk)k,O be an increasing sequence of positive real numbers such that (tk/tk+1)2< CO. Let A be the set of all finite where E, = 0 or 1. We shall prove (Chapter VIII, Secsums tion 5, Theorem IV) that, for each positive E,there exists a compact subset Ke of R such that sup If 1 2 (1 - E) [] f /Im for each almost periodic
FkaO
XkaO
+
K ,
function f whose spectrum lies in A. However, the sequence (tk)kSOcan be chosen in such a way that A is not harmonious (Chapter VIII, Theorem XI). 5. Pisot numbers and coherent sets of frequencies: the real case THEOREM IV: Let 8 be a real number greater than one andA, the set of all finite sumsxkgO&,Ok, where E, = 0 or 1. Then the following two properties of 8 are equivalent: (a) 8 is a Pisot number; (b) A, is a coherent set of frequencies. Before proving Theorem IV, some remarks should be made : if 8 = 1, A, is the set N of all positive integers and A, is a coherent set of frequencies; if 8 = - 1, A, = Z and is a coherent set of frequencies. If 8 < - 1 the proof given below will show that A, is a coherent set of frequencies if
111
and only if - 8 is a Pisot number. If - 1 < 8 < 1, Proposition 3 below shows that A, cannot be a coherent set of frequencies. Now to the proof. (a) Assume 8 is a Pisot number: then A, is harmonious and hence A, is a coherent set of frequencies; (b) Now assume 8 is not a Pisot number. The following lemma will be needed. LEMMA 6: If 8 is not a Pisot number there exists a sequence (Pk)k,O of Jinite trigonometric sums whose frequencies belong to Ae such that
(2) on each compact subset K of R not containing 0, P, tends uniformly to 0 as k tends to infinity. Proof of the lemma: Let Pk(t) = exp nit (1 + + 8,) cos n t cos nOkt; Pk(t) is a finite trigonometric sum of the type an exp 2ni;lt. If 8 is not a Pisot number, IP,(t)l = lcos ntj lcos nOktl is a decreasing sequence tending to 0 for each t # 0: if we write IcosnOkt12= 1 - sin2 n02t, this fact is a consequence of Pisot's theorem. Dini's theorem shows that (Pk)kaOtends uniformly to 0 on each compact set of real numbers not containing 0.
xnen
5.1. Proof of Theorem I V Assume that there exist a compact set K of real numbers and a positive C such that, for each trigonometric sum whose frequencies belong to A,, sup IP( < C sup IPI. Let s be a real number such that s K does not R
+
K
contain 0 and set Qk(t) = Pk(S + t). Then Q, is a finite trigonometric sum whose frequencies belong to A, sup JQkl-+ 0 as k + + co and K
IIQkllm= 1. Hence A, is not a coherent set of frequencies.
5.2. Improvement on the preceding result Let 8 be a Pisot number, q a real number in 10, 1[ and T(q) the infimum of those positive T for which sup IP(t)l 2 (1 - q) ll Pll for each ItkT
trigonometric sum P whose frequencies belong to A,. Two problems can be stated: what is the behaviour of T(q) as 11 t e ~ d s to O? What is the behaviour of T(q) as 7 tends to I ?
112
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
PROPOSITION 2: Let n be the degree of the Pisot number 8. There exist two positive constants a and b depending on 8, such that, for each q in 10, 1[, (-n+l)/z < , T(q) < bq(-nf1)/2. all This is the best possible result as Proposition 3 of Chapter I shows. This remarkable fact can be proved by the methods used in Chapter I, Section 4, and with the definition of models given in Chapter 11, Section 9. Observe that when q tends to 0, the quality of the prediction of the size of trigonometric sums whose frequencies belong to A increases at a corresponding 'cost', here the size of T(q), which is of the order of magnitude of v - ( " - ~ ) ' ~ . On the other hand when q tends to 1, the quality of the prediction decreases while the 'cost' of the prediction also decreases. What is the lower bound of this 'cost'? This is an open question. When q tends to 1, a natural conjecture is lim T(q) = DensA,. The density ofA, is defined V-+l
to be the uniform limit of T - l Card (A n [t, t + TI) as T tends to infinity. If 8 > 2 this density is 0, if 8 = 2 it is 1 and if 8 = (I + 4 5 1 2 (the golden number of Greek mathematicians) it is 3 + 3 &/lo. This conjecture will be provedin the case where Bis the quadratic Pisot number (1 + JT)/2 (Chapter V, Theorem 11).
113
Before proving Theorem IV, some remarks should be made :if 181, < 1, A, is an infinite subset of the unit ball of Q,. By Proposition 3 below, A, cannot be a coherent set of frequencies. We can therefore restrict our attention to those 8 in Q, such that 101, > 1. Now to the proof. (a) Assume 8 is a Pisot-Chabauty number: A, is harmonious and hence A, is a coherent set of frequencies; (b) Now assume 0 is not a Pisot-Chabauty number. The following lemma will be needed:
7.2. LEMMA 7: If 8 is not a Pisot-Chabauty number, there exists a sequence (P,),,, ofjinite trigonometric sums whosefrequencies belong to A, such that (1) sup lpkl = Pk(0) = 1 . QP
(2) on each compact subset K of Qp not containing 0, P, tends uniformly to 0 as k tends to infinity. Proof of the lemma: We recall that eachp-adic number can be written uniquely as a sum x = 5 r, where E E Z, and -3 < r < 3,r E A,, where A, is the subring of Q generated by p-'. We put r = p,(x) and the characters on Q, are of the form x -+ exp 2nipP(xA) for a suitable A in Q,. A finite trigonometric sum on Q, can be written a* exp 2nipp (AX). For each k 2 1, IetF, be the subset ofall sumsEk,-'~~8j, = 0 or 1 inA, andlet P, be the trigonometric sum2- '&,FK exp 2n@, (Ax). Then P,(x) can be written 2(1 exp 2nip, (Bjx)) and lPkl = Ilk-' ICOS n ~ (8jx)l. , Chabauty's theorem (Proposition 12, Section 12, Chapter 11) shows that if 8 is not a Pisot-Chaubauty number, P,(x) tends to zero as k tends to infinity for each x # 0. The proof is concluded as in the real case.
+
CAEap
6. A set of powers Let 8 be a real number greater than one and A the set of all powers Ok, k >/ 0. It will be proved that A is always a coherent set of frequencies (Chapter VI, Theorem 11). However, some problems remain open. Conjecture: Let 8 and A be as above. Then the following two properties are equivalent: (a) 8 is a Pisot or a Salem number; (b) for each positive q one can find a T(q) > 0 such that SUP Ip(t)l 2 (1 - 7) SUP Ip(t>l ItlG T ( q )
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
'nk-'
+
R
for each trigonometric sum P whose frequencies belong to A.
8. The groups C and R2
7. Pisot numbers and coherent sets of frequencies: the p-adic case
Complex Pisot numbers can be defined in the following way: a complex number 0 is a complex Pisot number if there exists an irreducible element P(X) = X n + a,Xn-' + ... + an of Z[X] with the following properties : (a) P(0) = P@) = 0 and 8 $- 8; (b) the other n - 2 roots z of P(z) = 0 satisfy lzl < 1.
7.1. THEOREM V: Let 8 be up-adic number andA, the set of allfinite sums Zk,oskBk, where E , = 0 or 1. Then the following two properties of 8 are equivalent: (a) 8 is a Pisot-Chabauty number; (b) A is a coherent set of frequencies in Q,.
114
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
It can be proved [43] that for a complex number 8, which is not real, the following two properties are equivalent: (a) 8 is a complex Pisot number; (b) 8 is neither purely real nor purely imaginary and the set A, of all finite sums ,, .skek,E~ = 0 or 1, is a coherent set of frequencies. Pisot pairs (a, 16) in R2 are defined in the following way: an element (a, p) in R2 is a Pisot pair if there is an irreducible element P(X) + a, of Z[X] with the following properties: = X" + a,Xn-l + (a) P(a) = P(P) = 0; (b) the other n - 2 roots z of P(z) = 0 satisfy lzl < 1. It can be proved [43] that for an element (a, p) of R2, the following two properties are equivalent : (a) (a,p) is a Pisot pair or la1 and 1161 are two Pisot numbers; (b) the set A& of all finite sums ( z k a OE ~ O L ~ ,ckpk)E~ = 0, 1 is a coherent set of frequencies.
xkZ
xkaO
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
LEMMA 8:
If y ET,y' E
and y # y', then sup JxY(x)- xy.(x)l > xeG
115
43.
Proof: Let 6 = y - y'; then, for each x in G, (xY(x)- xY,(x)( I~a(x)- 1I. Since 6 # 0, the subgroup x,(G) of T is not the trivial subgroup (1). Hence there is an x in G such that Jx,(x) - 11 > J3. Now to the proof of Proposition 3. Let V be a neighborhocd of 0 in such that sup Ix,(x) - 1I < 1/(C 43). If S = y - y' E V, we have =
r
KxV
sup I~a(x)- 11 = sup Ixy,(x) - xy(x)I > l / ( J 3 C), which contradicts the K
K
definition of V. We shall say that A is uniformly discrete to express the conclusion of Proposition 3. If G is discrete, 7 is compact and each coherent set of frequencies A contained in T is finite. Hence coherent sets of frequencies are interesting when G is neither compact nor discrete.
10. Coherent sets of frequencies and restriction algebras
9. Coherent sets of frequencies in 1.c.a. groups The definition of a coherent set of frequencies will be given for the case of a general 1.c.a. group. DEFINITION 4: Let G be a 1.c.a. group, T the dual group of G and A a subset of A is a coherent set of frequencies if there exist a positive real number C and a compact subset K of G such that, for each trigonometric sum P(x) = xlen c r ~ ~ ( we x ) have
r.
(1) sup IPI G
< Csup IPI. K
If(1) holds we say that the pair (K, C) is suitable for A. Example: If G is compact, each subset A of has this ploperty. In the general case, each harmonious subset A of T has this property. But there seems to be no hope, even if A is relatively dense in of obtaining a characterization ofA as we did for harmonious sets.
r
r,
PROPOSITION 3 : LetA c I' be a coherent set of frequencies. There exists a neighborhood V of 0 in such that, if 2 EA, A' EA and 2 f: A', then 2' - A 4 v. The proof depends on the following lemma:
r
10.1. For each closed subset A of r , two Banach algebras A(A) and B(A) can be constructed: A(A) is the quotient algebra of the group algebra L1(G) by the closed ideal I(A) of L1(G) defined by f E I(A) if its Fourier transform vanishes on A ; B(A) is defined in the same way with the measure algebra M(G) in place of L1(G). We have the following surprising characterization of coherent sets of frequencies: A is a coherent set of frequencies if and only ifA is discrete and B(A) is the multiplier algebra of A(A). We first give some new definitions.
r,
DEFINITION 5 : Let E be a closed subset of the dual group of G. The Banach algebra A(E) is dejined by the following two conditions: (1) p, belongs to A(E) if there exists an f E L1(G) such that p, = f o n E; (2) llv/IA(E) = inf 11 f Ill, where the infimum is taken with respect to those f E L1(G), such that p, = .f on E. DEFINITION 6: Let E be a closed subset ofr. The Banach algebra B(E) is dejined by the following two conditions: (3) p, belongs to B(E) if there exists a bounded complex-Radon measure ,u E M(G) such that p, = , ii on E; (4) IIpIIBcE>= inf Ilpil, where llpll is the 'total variation' of p. DEFINITION 7: Let E be a closed subset of A P (G) is defined by
T.The subspace APE (G) of
116
PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS
PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS
(5) f E APE (G) if the spectrum of the almost-periodic function f is contained in E and the subspace L,"(G) is defned by (6) f E Lz(G) if the spectrum of the bounded Lebesgue-measurablefunction p is contained in E. Remarks: The Banach spaces APE (G) and B(E) can be put in duality. I f f E APE (G) is a finite trigonometric sum f(x) = CysE ayXy(x) and if p = ,L on E, we define n
117
(b) (a): The proof depends on the following construction of linear forms over the space APA (G). Let x E Hom (r,, T) be any homomorphism from the additive group I' (with the discrete topology) to T; for each f E A P (G), f(x) is defined by the following rule: iff is a finite trigonometric sum f(x) = zy,ayX, (x), f(x) is the corresponding ~ ~ m ~ ~ Kronecker's theorem shows that, on the finite spectrum A off, the complex numbers ~ ( y )y, EA, can be approximated by restrictions to A of continuous characters on T ; hence (f(x)I < sup If 1 and the linear form G
f +f(x) can be extended by continuity to the whole of AP (G). The following lemma will now be used:
We get I(f, p)l < Il f 11, llvllBcE,and, by Theorem I, the linear form f + (f, p) can be extended to the whole of APE (G).
10.2. Duality and coherent sets of frequencies It may be asked whether B(E) is the dual space of APE (G). The answer is surprising: THEOREM VI: Let G be a metrizable and separable 1.c.a. group andA a closed subset of the dual group F. The following two properties of A are equivalent: (a) A is a coherent set of frequencies; (b) each continuous linear form L on the space of all almost periodic functions P whose spectra lie in A can be written L(P) = f G P(x) dp (x).for a suitable complex valued bounded Radon measure p on G. Before proving Theorem VI some remarks should b: made. If for each positive 7 in 10, 1[ there exists a compact subset K of G such that the pair (K, 7) is suitable forA, then the dual space of APA(G) is isometric to B(A), as the following proof shows. In all other cases, the isomorphism between B(A) and the dual space of APn (G) is not an isometry. 10.3. Proof of Theorem VZ
A coherent set of frequenciesn is discrete in r;hence A is closed. (a) * (b) : Let (K, C) be suitable for A and let 8 be the closed subspace of %(K), whose elements are the restrictions g to K of all f in APA(G). A continuous linear form I on APA(G) defines a continuous linear form L on 6. The Hahn-Banach theorem shows that there is a measure p carried by K such that, for each f E APA(G), l(f) = f G f dp and (b) follows. The norm of p satisfies the inequality llpll < Clllll.
LEMMA 9: Let G be a metrizable and separable 1.c.a. group, r the dual group andA a subset ofr. Assume thatA is not a coherent set of frequencies. Then for each positive E, there exist a sequence (Pk)k>Oof trigonometric sums whose frequencies belong to A and a x in Hom (T,, T) such that (a) sup IPkl = 1 ; G
(b) on each compact subset K of G, sup lPkl+ 0, as k + +a; K
(c) IPk(x)I > 1 - E for each k 2 0. Before proving the lemma, we show why it implies Theorem VI. Assume that the linear form f +f(x), defined on APA (G) by the element x of Hom (T,, T) found in Lemma 9, can be written f (x) = fGf ( - x) dp (x) with p E M(G). Replacing f by Pkand using Lebesgue's dominated convergence theorem, we get f G P,(-x) dp (x) + 0, while IPk(x)I > 1 - &. This is impossible and Theorem VI is proved.
10.4. We return to the proof of Lemma 9. Let (K,),,, be an increasing sequence of compact subsets of G whose union is the whole group G. A sequence (Pk)k>Oof trigonometric sums on G whose frequencies belong to A and a sequence (xk)kgOof elements of G will be constructed by induction in such a way that sup lPk(= 1 G
(4.5)
~
~
118
PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
We assume that x , , ..., x,, P,, ..., P, have been chosen and we now define x,,, and P,, , noting that the choice of x , and P, is easy. Since A is not a coherent set of frequencies, there exists a sequence ( Q J k a oof trigonometric sums whose frequencies belong to A such that sup IQ,J = 1 a r d sup lQ,l + 0 as k -> +a.The trigonometric sum
,
G
Kk
+ P, is an almost periodic function, as is evzry trigonoS, = P, + metric sum; so is Re [S,] and there exists a compact subset L , of G such that each translate s + L , contains an x satisfying Re [S,(x)] 3 sup Re [S,] - ~ 2 - ~ -3' k - F (2-I + + 2 - , + 2-k-2). G
Let m = m , be a sufficiently large integer that IQrn(x)I< l / ( k + 1 ) on K,,, - L,. Now regarding m as fixed, there exists an element s = s, in G such that IQm(s,)I > 1 We can multiply Q, by a constant complex number of absolute value 1 , which does not affect the other pro> 1 - d-,-'. NOWs, + Lk contains an perties of Q,, to get Re [Qm(sk)] element x k + , of G such that
119
11. Other definitions of coherent sets of frequencies
11.1. We recall that Hom (A,, T ) is the group of restrictions to A of all homomorphisms of the discretized group r i n t o the circle group T . Then thefollowing two properties of a closedsubsetA of r a r e equivalent: (a) A is a coherent set of frequencies; (b) Hom (A,, T ) is contained in B(A). Let A be a coherent set of frequencies. Each x E Hom (A,, T ) defines a linear form f +f ( ~ )on APA (G) and there is a ,u E M(G) such that f01> = jGf ( - X ) dp ( x ); this implies &(A) = ~ ( 1for) each A in A. If A is not a coherent set of frequencies, Lemma 9 gives a x E Hom (A,, T ) , which is not in B(A). 11.2. Let A be a closed subset of r.We say that a function y : A + C is a multiplier of A(A) if for each f E A(A), yf (the usual product) also lies in A(A).
r,
Putting P,+,(x)
=
Q,, ( x - x,,,
THEOREM V I I : Ij2l is a discrete subset of the following two properties of A are equivalent: (a) A is a coherent set of frequencies; (b) the multiplier space of A(A) is B(A). The proof of this result depends on the following interesting lemma:
+ s,), we have
and
The induction is thus complete. The inequalities (4.7) imply IPj(xk)l 2 1 - E for each k > 0 and each j less than k. Following Section 1 of Chapter 11, we regard G as a part of G" = Hom ( r d ,T ) ; let C , be the closure in G" of the tail-set {x,, x,, , .. .). Sii?ce G" is compact, there exists a 31 in G" belonging to all the C , , k > 0. Using the continuity of Pj for each fixed j, we get IP,(x)I 3 1 - F ; since x is independent of j, Lemma 9 is proved.
,
LEMMA 10: Let A be a discrete subset of a 1.c.a.grouprandlet f : A + C be afunction such that f(A) = 0 for all A in A with the exception of at most a cAf @)I,this supremum jinite number of A E A . Then Il f IIA(A) = sup being taken over all Jinite complex sequences (c,),,, such that P(x) = C n X A ( x ) satisjie~llPllrn < 1. Proof of Lemma 10: Let a = 11 f llAcA, and let b the right-hand side of the hoped-for equality. The inequality b < a is obvious. For each F > 0, there is an F in L1(G) such that llFlll < a F and P = f on A. Hence c ~ f (= 4 JG P( -x)F(x)dximpIie~ l & e ~ c ~( f4 < I1 PII, IIFII 1 < a + F. Thus b < a + E for all positive F , which grves b < a. We want to get the reverse inequality. Let E be any positive real number and g an element ofL1(G) with the following three properties: $(A) = 1 if A E A and f(A) # 0 , g has compact support and llgll < 1 F ([15],p. 53, Theorem 2.6.8). Let E be the finite set of all A in A for which $(A) Z 0. Theorem V I applied to the harmonious set E gives 11 f l l ~ ( ~ 6 ( I n a o lc(2")I). This would imply 8 Ic(2")I) < (c + 1) Il f lla for any n/5-periodic function. But any sequence (x,).,~ in lZis the sequence of Fourier coefficients 42") of a n/5periodic function ([24], th. 7.1, ch. XII).
+
-
xzs
En,
-
(En,,
131
(a) iff is a complex-valued bounded continuousfunction whose spectrum lies in A + E, g = L(f ) is an almost-periodicfunction whose spectrum lies inA F ; (b) sup lgl < C sup 1 f 1; moreover there is a real continuous function w
+
G
G
on G, vanishing at injinity, such that sup Igl G
(c) for each x in G, Ig(x) - f(x)l
< 1x1 sup bf I;
< sup lwf 1; G
< Clxl sup G If 1 ;moreover Ig(x) - f(x)l
G
(d) for each 2 in A, if the spectrum off lies in 2 + E, the spectrum of g lies in 2 + F. Before proving Theorem XI, a remark should be made: since F is finite, the almost-periodic functions whose spectra lie in A + F are in a class of almost-periodic function because A + F is harmonious. Roughly speaking, the aim of theorem XI is to replace a bad function f by a regular one g without serious modification of the spectrum.
15.6. Summary of the proof The idea of the proof is to consider T as a dense subgroup of the Bohr compactification f of and, using a partition of unity on 17 by means of the regular functions, to divide A into a finite number of pieces AT for which Proposition 9 applied to the pair (E, A,*) will be true. We recall that the Bohr compactification f of is the dual group of Gd, where Gd is G with the discrete topology; A(P) - is the Banach algebra of complex valued continuous functions y on whose Fourier series are absolutely convergent. A continuous homomorphism h : f with a dense range is defined by the following rule: for each y ET,h(y) is the character X, on G. Then, if y E y 0 h is an almost periodic function on I' whose Fourier series is absolutely convergent.
r
r
15.5. A fundamental theorem With a very slight modification of the range of the operator L, the preceding difficulty disappears completely. Roughly speaking, Theorem XI states that each slowly-perturbed f,(x) x,(x) can be approximated by a pure trigonotrigonometric sum P,(x) x,(x), where the frequencies of each trigonometric metric sum sum PAbelong to a fixed finite subset F of the compact set E containing the spectrum of each f,. (We have introduced new frequencies to get a better approximation; A is harmonious and no restrictions have to be made on E.)
FA,, Inon
r
THEOREM XI: Let G be a separable and metrizable 1.c.a. group, the dual group andA a harmonious subset of T. For each compact subset E of I', there exist aJinite subset F of E, a positive constant C and a linear operator L, whose domain and range will be specijied in a moment such that
r
r-t
~(0,
15.7. A set F , If A is harmonious in I',let M(E), for each positive E, be the relatively dense subset of G defined by sup Ixn(x) - 1I < E. Let A, be the set of all A
y in
r such that Ixy(x) - I I < 28 for each positive E and each x in M(E).
Then A, is harmonious by Theorem I of Chapter I1 and (E - E) n A, = F , is finite. It is clear that F , = - F , and 0 E F1.Finally let be the set of all y in I' such that IxY(x)- 1I d s for all positive E and each x in
132
PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS
PROBLEMS O N A-PERIODIC TRIGONOMETRIC SUMS
M ( E ) ;A is harmonious by Theorem I of Chapter 11. The above definition of F , seems artificial but Fl plays a fundamental role when A is mapped into the Bohr compactification f of I'. Bohr compactijication, U, A*, U*. Let h : I'-* 1; be the homomorphic injection of I' in f and let U be the closure of h(A) in f. The inverse image under h of the compact subset U o f f is again an harmonious subset of I' contained in as the following observation shows: For eachpositive E , each t E M ( E )and each x E U , we have Ix(t) - 1 I < E. The inequalities Ix(t) - I I < E , where t runs over M ( E )define a closed subset o f f containing h ( ~and ) hence containing U. For each y in h-'(U) we have Jxy(t)- 11 < E ( t E M(E))and h-'(U) c A. Let V be the compact subset h(E) o f f . The following lemma shows that the number of decompositions of an element x of f as a sum 5 g, 5 E U , g E V is finite; the finite set concerned here is precisely F,.
Ix(t) - g(t)l < & (resp. Ix(t) - g(t)l < &) for each t in T . Finally let cc, be the set of all g in f such that Ir(t) - 1) < for all t E T. As x runs over U , the family consisting of the corresponding a, is an open covering of U. There exist an integer n and a finite covering of U by o , , ..., on, where oj = w, for x = xj E U, 1 < j < n. We define U* = U + h(Fl), Uj* = U* n w,*, where a,*= w,* for x = xj and o,*is a compact subset of f'. Finally let A,* be the set of all A in I' such that h(A) E Uj*.The set A,* is contained in A* and the union U AS contains A.
+
+
LEMMA 13: If an element x of 1; can be written x = 6 q = 5' + qf, where 5 and 5' belong to U and g and g' belong to V, then 6 - 5' = g' - g belongs to h(Fl). Proof: By the definition of V, we have g' - g = h(y),where y E E - E. On the other hand, for each t E M(E),1Qt) - 11 < E and I['(t) - 11 < s, which imply I([ - 5') ( t ) - 11 < 28. Then IxY(t)- 11 < 2s (t E M(E))and y E E - E show that y E F 1 , since this is the definition of F, . In what follows, the compact subset U* = U + h(Fl) plays an important role; h-l(U*) = h-l(U) + F, is contained in the harmonious set A* = A -t F , (Chapter 11, Section 3, Theorem 11).
l<j l< clxl Ilfllm; (5) if the spectrum off is contained in A* E l , A* E Aj*, the spectrum of Lj,k(f) is A* sk; (6) there is a real continuous function q,, on G vanishing at infinity such that IILj,k(f)II rn < I l o j , k f ll m . We now consider an f whose spectrum is contained in A + E. We define J;., 1 < j < n, by J;. = f * oj; the spectrum of fj is a subset of (4) I(Lj,k(f>>
+
+
136
137
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
PROBLEMS ON A-PERIODIC TRIGONOMETRIC SUMS
A: + F, + E, which is contained in the interior of A,* + El. For this f,,, and hence reason we can define&,, to bef, * pjSk.We have f, =
(a) (b): If 8 is a Pisot number, A, is harmonious and Theorem XI1 is a weak version of theorem XI. (b) +-(a): The proof is based on the existence of a compact set E of real numbers with the following properties: (1) the vector space X generated by E over Q has Lebesgue measure 0; (2) 0 belongs to E; (3) there exists a complex valued continuous function pl on R vanishing at infinity, whose spectrum is contained in E and such that ~ ( 0 = ) 1. The existence of such a set will be proved in Chapter VIII, Section 10.
z
l (b); A is a relatively dense harmonious subset of r:for some compact subset K of r , A + K = r. Theorem I1 of Chapter IV shows that for each positive E there is a relatively dense set of &-almost periods for all f in E(A). The proof of the last statement of (b) is an easy adaptation of the proof of part (b) in Theorem IX of Chapter V. For this reason, the reader is referred to Chapter V, Section 8.7. Proof of (b) =. (a): The existence, for each positive s, of an a priori complete set M(E) of &-almostperiods for all f in E(A) implies that A is harmonious. To prove that A is relatively dense, from which (a) will follow, we use the following lemma :
INTERLUDE
One of the most fascinating problems in harmonic analysis is to find connections between the additive properties of a set A of frequencies and the functional properties of the corresponding space of almost periodic functions whose spectra are contained in A. ForA a finite set of real numbers independent over Q, we obtained the following connection in Chapter I: we can rapidly predict the size of trigonometric sums whose frequencies belong to A if and only if A is strongly independent over Z (Chapter I, Theorem IX). In the general context of 1.c.a. groups, we can also get striking results. We first recall some definitions: (1) A subset A of a 1.c.a. group G is discrete if each point 1 of A is isolated in A for the relative topology; (2) A subset A of G is relatively dense if there exists a compact subset K of G such that each translate of K intersects A ; (3) A set E of bounded continuous functionsf : G -+ C has a complete set of a priori &-almostperiods if there exists a relatively dense subset M(E) of G such that, for each z in M(E) and each f in E, sup f ( x + z) G - f ( x ) l Q E SUP If I ;
LEMMA 1 : For each n > 1, let A, be the set of a l l y in r whose distance from 0 exceeds n. Then for any h :I' -+ C in A(r), the norm of h in the restriction algebra A(A,) tends to 0. Proof: Obvious, since the compactly supported h in A(r) are dense in A(r). We return to the implication (b) (a). If E(A) is trivial on a compact neighbourhood V of 0 in G, a theorem of Banach applied to the surjective restriction operator T: E(A) -+ C(V) yields a constant C such that, for each complex valued continuous g, in C(V), there exists an f in E(A) such that sup If 1 Q C sup Ig,I and f lv = 9.
I
G
(4) A set E of bounded continuous functions f : G -+ C is locally trivial at 0 if there exists a compact neighbourhood V of 0 such that each continuous function g : V 4 C is the restriction to V of a function f in E. In this case we say that E is trivial on V. With these definitions we can state the following: THEOREM I: Let G be a metrizable and separable 1.c.a. group, I'the dual group, A a subset of I', and E(A) the space of all almost periodic functions f:G -+ C whose spectra lie in A. The following two properties are equivalent: (a) A is discrete, A is relatively dense in P and there is ajinite subset F of r such that A - A is contained in A F ; (b) for each positive s, there is a complete set of a priori s-almost periods for all f in E(A) and E(A) is locally trivial. Before giving the proof, some examples should be noted.
+
141
G
,'
V
It follows from duality that for each complex valued Radon measure p carried by V and each bounded complex valued Radon measure v, the equalities $(A) = $(A) for all 2 EA imply Ilpll < Cllvll. Let h be the Fourier-transform of an integrable function I, not identically zero, which is carried by V; then h E A(r). IfA is not relatively dense in G, there is for each n > 1, a yn in r whose &stance from A exceeds n. Let dp, = y,(x) I(x) dx. Lemma 1 shows that there is a sequence v, of complex Radon measures such that llvnll -+ 0 and ,il,(l)= Jn(2) for all A inA. But IIpnll = 111111and this contradicts llpnll Q Cllvnll.
SPECIAL SERIES
CHAPTER V
that, for every a
E 9'
10 SPECIAL T R I G O N O M E T R I C SERIES (COMPLEX METHODS)
(COMPLEX METHODS)
satisfying
+ 1x1)" y'k'(x)l < C,
0
< k < n,
(5.1)
we have IT(a)l < 1. The Fourier transformation a + & is an isomorphism of Y and the Fourier-transform of each T i n 9' is defined by the rule ( T , &) = (F, a ) . An element T of Y' is supported by [0, oo[ if for each a E Y which is 0 in a neighbourhood of [0, a [ , we have ( T , a ) = 0. From now on we shall c o n h e our attention to those T E Y' supported by [0, + a ] .
+
In the real and the p-adic case, precise and interesting results about coherent sets of frequencies need complex methods. In Section 1 we define the Laplace or the complex Fourier-transform of a distribution with compact support. The Paley-Wiener theorem gives precise information on the set of zeros of such a transform (Sections 2 and 3). In Section 9, we present the theory of mean-periodic functions on the line. Studying coherent sets of frequencies is equivalent to investigating the behaviour at infinity of all mean periodic functions whose spectrum is a given set of real numbers. Trigonometric series whose frequencies belong to the set of all entire parts of kB, where k 2 1 and 13 is an irrational number greater than one have quite fascinating properties which are studied in Section 6. But perhaps the most beautiful results concern the trigonometric series al exp 2ni2x, where 2 = mpn, defined on thep-adic field Q, by f ( x ) = CAE, for rational integers m and n such that 0 < mpn < 1. These series have almost all the properties of 2mperiodic functions on the line and other better ones (Section 8). The harmonious set A and its homothetical copies pkA, k 2 1 , are of outstanding importance in the atomization of distributions in Q,, just as arithmetic progressions are used to discretize problems on the real line.
143
+
1.2. The domain of dejnition of the Laplace transform
+
Let p = u iu be a complex number. The function t -+ e-"' does not belong to 9 , but i f a is an indefinitely diflerentiable function which is 1 on [- 1 , co[ and 0 on ] - oo, -21, then a ( t ) e-"' belongs to Y for each p E C such that u = Wep > 0. We can define ( T , e-"') to be ( T , a ( t ) e-P') whenever T is supported by [0, a [ . From now on, we write e-"' T ( t ) dt instead of ( T , e-"'). The Laplace transform can be defined on a larger space of distributions. For instance consider the function exp J7, t 2 0. For each p E C whose real part is positive, the integral j2 exp ( - p t &) dt converges and defines the Laplace transform of this function. Let 9 be the space of all complex valued indefinitely differentiable functions on the line with compact support. A distribution is a linear form T over 9 such that for each m Z 1 there is an integer n > 0 and a constant C > 0 with the property that for all y in 9 supported by [ - m , m ] the inequalities sup Ip(j)(t)J< C, 0 < j < n, imply IT(y)l < 1.
+
+
+
1-m,ml
If this is the case we write T E 9'. The domain of definition of the Laplace transform is the subspace 9of 9' consisting of all distributions T carried by [0, oo] such that, for each E > 0, e-et Tbelongs to 9'.
+
1. The Laplace transform 1.1. The space of tempered distributions
Following the notation of [21], Y is the Frechet space of all complex valued functions a on R all of whose derivatives, together with a itself, decay rapidly at infinity: for each j Z 0 and each k 2- 0, (1 Ixl)j y'k' ( x ) + 0 as 1x1 + + oo. The dual space Y' is the space of tempered distributions. For each T E Y' there is an integer n and a constant C > 0 such
+
142
1.3. Properties of the Laplace transform PROPOSITION 1 :For all T i n 9,theLaplace transform @(p)= :j e-"'T(t) dt is holomorphic in the open half-plane Wep > 0. Proof: Let 0 < E < Wep. Then e-pr = e-"e-'p-e'r , e-er T ( t ) E 9' and W e ( p - E ) > 0. The integral is thus well defined.
S E 9"; if S E ~we, apply the preceding argument to S ( t ) exp ( - s t / 2 ) with s/2 instead of E. The first part of Theorem I is thus proved. In the second half of Theorem I, the unknown distribution Twill be a limit of convolutions T * f as f runs over a sequence f , of indefinitely differentiable functions of the form f k ( t ) = k g ( k t ) , where g is supported by [O, 11 and j g(t) dt = 1. From now on, we write f instead off,. Let F(p) be the Laplace transform off. The Laplace transform of ~ k f is pkF(p). Now the Laplace transform of a complex valued continuous function carried by a compact subset of [0, + a [ is bounded. Hence 1pkF(p)l < c k .
To prove that @(p)is holomorphic, we apply the same trick and we are thereby reduced to examining only the case where T belongs to 9". If Wep, > 0, then with a defined as above, ~ ( t(exp ) ( - p t ) - exp (-pot)/ p - po tends to -ta ( t ) exp ( - p o t ) in the topology of 9'.Hence @(p)is differentiable over the field C at the point po and @'(p) is the Laplace transform of - t T (t). THEOREM I : Let T be a distribution carried by [0,+ co[ such that, for each positive E , e-,'T is a tempered distribution. Then the Laplace transform @(p)of T is a holomorphic function in the open halfplane with the following property: there is an integer n 2 0 such thatfor each positive s, we can find a constant C , such that (5.2) I@(p)l < C,lpln for Bep 2 s. Conversely each holomorphic function in the half-plane 9 e p > 0 satisfying (5.2) is the Laplace transform of a unique distribution T carried by [0,i-a[ such that e-" T belongs to 9" for each positive E. The proof of the first half of Theorem I begins with a lemma; here T is any distribution carried by [0, a[.
91ep20
The p~oduct@,(p) = F(p) @(p)has the following behaviour at infinity: for each positive E > 0 and each n > 0, Idil(p)l < C ( n , E ) IpI-" when Wep 2 E.
+
+
LEMMA 1: There exist an integer n >, 0, a continuous complex oalued function f carried by a compact subset of [0, + a [ and a distribution S carried by [l, + co [ such that T = D n ( f ) + S where Dn is the nth (distributional) derivative. Proof of the lemma: Let B be any indefinitely differentiable function with compact support, which is equal to 1 on a neighbourhood of [0,11. We write T = BT + ( 1 - B) T . The distribution /IT is carried by a compact set. Hence there exist an integer n 2 0 and a continuous function g such that g(t) = 0 if t < 0, g(t) = c if t >, to and PT = Dn(g).Let y be any indefinitely differentiable function with compact support, which is equal to 1 on a neighbourhood of [0,t o ] We define f = yg; then D n ( f ) = Dn(g) on a neighbourhood of [0,to] and T = D n ( f ) + S is the required decomposition of T. Proof of the first part of Theorem I: Let @, be the Laplace transform off and@, the Laplace transform of S. Then @, is an entire function ofp, bounded on Bep 2 0. The Laplace transform of D n ( f ) ispn@,(p)and satisfies (5.2). On the other hand, let a be an indefinitely differentiable function which is 0 if t < and 1 if t 2 2. Then for each s > 0 and each n 2 0, there is a constant c = c (n, E ) such that each function q ( t ) = ~ ( texp ) -pt (Wep >, E ) satisfies (5.1). This implies I<S, q)l < C, if
+
LEMMA 2 : For each real t , the integral ( 1 1 2 4 t): @, ( u iv) et("+'")dv T f ( t )exists and is independent of u > 0.This integral vanishes i f t < 0. Proof of the lemma: Let u, < u, be two such u. We apply Cauchy's theorem on the rectangle whose vertices are (u, , -L), (u, , -L), (u, ,L ) and ( u l , L). We get =
i
i StL@, (u,
+ iv) ef(u"i") dv - i j!,
Dl (u,
+ iv) e"U1'i"' dv
The behaviour of @, at infinity ensures that the last two integrals tend to 0 as L tends to infinity. I f t < 0 and u 2 1 , IJ+; @, (u + iv) et(u+iu)do] < C ( 2 , 1) j?," lu ivl-, dv, which tends obviously to 0 as u tends to + a .
+
LEMMA 3 : For eachjixed u > 0,e-'"Tf ( t ) E Y . The Laplace transform of T f is @, . Proof: The function e-'"Tf ( t ) is the inverse Fourier transform of @, ( u iv). To get estimates of the derivatives of @, , we apply Cauchy's formulas on the rectangle with vertices ( 4 2 , v/2), (3u / 2 , ~ / 2 )( ,3 ~ 1 23012) , and ( 4 2 , 3 ~ 1 2 ) :the inequalities I@, (u/2 iv)l < C'(u, n) Ivl-" and I@, ( 3 ~ 1 2 iu)l < CV(u,n) Ivl-" imply that, for each fixed u, each derivative of @, (u iv) with respect to v decays rapidly at infinity. Hence v -+ @, ( u iv) belongs to Y and taking the inverse Fourier transform,
+
+ + +
+
The Fourier-transform of edtUT f ( t )is ( u + iu); the Laplace transform of T , is therefore @, . We now replace f by f,(t) = k g ( k t ) and we write T k instead of Tf,. LEMMA 4: AS k tends to inJinity, T k tends to a distribution T carried by [O, + co [ in the sense that for each indeJinitely diferentiable function h with compact support, ( T k ,h ) + ( T , h ) as k -+ + co. Proof: Let E be any positive real number. The Fourier transform of e - , ' ~ , is @, ( E + iv) = @ ( E iv) yk( E + iv), where yk is the Laplace transform of k g (kt). We remark that lYkl < A throughout the halfplane Weu >/ 0 for some A independent of k and that Yk-+ 1 uniformly on compact subsets of this half-plane. The function eeth ( t ) is the Fourier transform of a function r(v) in Y . @ ( E + iv) Y k ( &+ iv)r(v)dv Hence ( T k , h ) = ( e - " ~ ~ eeth) ,, = + ST: @ ( E + iv) r(v) dv as k + co (we have I@ ( E iu)I lr(v)I dv < + co since @ grows slowly at infinity and r decays rapidly at infinity and Lebesgue's dominated convergence theorem shows that the limit exists). If a distribution T is defined on R by the statement that the inverse Fourier transform of @ ( s + iv) is e-" T ( t ) , T would appear to depend on E. But the preceding argument gives, lim ( T k , h ) = ( T , h). Hence T k++m does not depend on E , Tis carried by [0, + oo [ and the Laplace transform of T is @.
+
+
(b) there exist an integer n 2 0 and a constant C such that IF (iy)l < C (1 lyl)". Conversely each entire function F(z) satisfying (a) and (b) is the Laplace transform of a distribution carried by [-I, I].
+
2.2. Proof of the first part of Theorem 11 If S is carried by [-I, I ] , we can find, for each positive E, two continuous functions f and g carried by [-I, I + E ] and an integer n 2 0 such that S = D n ( f ) + g. The proof is the same as that of Lemma 1. Let @(z) be the Laplace transform off, G that of g. Then F(z) = zn@( z ) + G(z). @(z) = j'_:,exp ( - z t ) f ( t ) dt implies I@(z)l < j'_:'exp ( - x t ) I f(t)l dt < C(E)exp (1 + E )1x1. The same inequality holds for G(z). Hence
+
2. The Paley-Wiener theorem
2.1. We list a number of corollaries of theorem I which are interesting in themselves. But first a definition is needed: 1: A complex valued.function f dejined in the complex plane DEFINITION is of exponential type I if, for each positive E , there exists a constant C(E) such that I f(z)l < C(s) exp [(I + E )lzl] for all z. A similar definition can be given for functions of a real variable. We say that f is of exponential type iff is of exponential type I for some I in 10, co [.
+
THEOREM 11: Let S be a distribution carried by [-I, I]. Then the Laplace transform F(z) of S is an entire function in the complex plane with the following two properties: (a) F is of exponential type I;
which implies (a) and (b). 2.3. To prove the converse, let F(z) be an entire function satisfying the inequality (5.3) for each positive E. Then exp [-(I + E )z ] F (z) is holomorphic in 9 e z > 0 and is bounded by C(E)(1 + IzIn). Theorem I shows that exp [-(I + E )z ]F ( z ) is the Laplace transform of a distribution T carried by [0, + co [. Hence F(z) is the Laplace transform of a distribution S carried by [- 1 - s, a[.In the same way, it can be proved that S is carried by ] - oo,I + E ] . The uniqueness of S shows that it is carried by
+
[-I,
4.
It remains to show that (a) and (b) imply (5.3). We use the following lemma of Phragmen-Lindelof : LEMMA 5: Let Y be a closed sector in the plane whose opening is less than n radians, and let f ( z ) be an holomorphic function on a neighbourhood of Y. I f f is of exponential type on Y and if( f(z)l < 1 on the frontier of Y, then ( f1 < 1 on the whole of Y. Proof: Since this statement is invariant under rotation, we can assume that Y is defined by -a < Arg z < a < 4 2 . Iff tends to 0 when z E Y and lz] -+ co, the maximum modulus principle gives I f(z)l < 1 for all z in Y. If f ( z ) does not tend to zero at infinity, we apply the preceding argument to f,(z) = exp (-EZ") f(z), where a > 1 and a a < 4 2 . We have lexp ( - E Z " ) ~= exp ( - E W e (za))< exp ( - s lzla cos aa). Hence I f,(z)l < 1 for each frontier point of Y andf , tends to zero when z E Yand lzI -, + co.
+
The preceding argument gives I fE(z)l < 1 for all positive E and all z in Y. Letting E tend to 0 , we get I f(z)l < 1. COROLLARY 1 : I f F is holomorphic in a neighbourhood of x 2 0,y 2 0 , i f F is of exponential type and ifJF(x)l < C exp Ix, IF (iy)l < C (1 (yl)", then (F ( x iy)l < C (1 Jzl)"exp lx. Proof: We apply Lemma 5 to f ( z ) = C-l ( z + i)-" F(z) exp -1z. The proof of Paley-Wiener's theorem is complete. We have proved the following improvement of Paley-Wiener's theorem. If F is an entire function of exponential type (not specified), if there exist an integer n 2 0 and a constant A such that IF (iy)l < A (1 Jyl)",and if for some positive I there is for each positive 8 a constant C , such that IF(x)J < C , exp (I + E ) 1x1, then F is the Laplace transform of a distribution carried by [-I, I].
+
+
n(r) = k nl(r). Once Theorem IS1 is proved for F , ,this relation at once gives the result for F. We apply Jensen's formula to F , to get that for each positive E and each R 2 1
+
+
+
Since this inequality holds for each positive E , Theorem I11 is proved. Remark: If n(r)/r tends to a limit D as r tends to infinity, so does R - l n(r)/r dr and Theorem 111 gives D < 1. In general, let D = l& n(r)/r. Then i E i R-l n(r)/r dr 2 D r-r + m
and Theorem 111 gives D
3. Repartition of roots of entire function of exponential type In all of what follows, F will be an entire function of exponential type I, so that for each positive E , there is a constant A(&) such that JF(z)l < A(&)exp (1 + E )(zI for any complex number z. Let a,, ...,a,, . . . be the sequence of all roots of F, ordered in such a way that la,l < la,+,l, and with each root occurring in the sequence (a,),, a number of times equal to its multiplicity.
,
DEFINITION 2: For each positiue real number r, n(r) is the number of roots of F in the closed disc lzl < r; alternatively, n(r) is the sup of n > 1 such that la,] < r. The function n(r) is defined on [0, +a[, is integervalued and is a step function. THEOREM 111: If E is an entire function of exponential type I, then lim sup R-I 1: n(r)/r dr < 1. We start with the well-known Jensen's formula: if F(z) is holomorphic on a neighbourhood of Jzl < R and if 0 < [all < la21 < ... < la,l is the sequence of roots of F, then dr o
r
=
S:'
2n
log IF ( ~ e ' ~do. )l
The proof is given in [16],th. 4.1, p. 181. We return to Theorem 111. I f F ( 0 ) = 0 , F(z) = zkF, (z),where F,(O)# 0 and if nl(r) is the number of roots of F , in the closed disc lzl < r, we have
r
o
Hence
< 1.
R-+m
,,,
COROLLARY 2: Let (A,)-, +, be an increasing sequence of real numbers such that lim I k / k = I / D exists. If 1 < n D , ecery distribution S Ikl-r+m
carried by [ - I , I ] whose Fourier transform canishes at all the A,, k is the zero distribution. Proof: Apply the Paley-Wiener theorem and Theorem 111.
E
Z
4. Bernstein's inequality THEOREM IV: Let g, be a complex valued bounded continuous function on the line whose spectrum1 Iies in [ - I , 11, or equivalently, a complex valued bounded continuous function on the line which is the restriction to R of an entire function of exponential type I. Then sup 19'1 < I sup Jq1. R
R
In particular for each pair of real numbers x' and xu, Ipl(x') - y(xV)l < 2 Ix" - x'l sup 191. R
Proof: The equivalence of the two definitions of is a consequence of Paley-Wiener's theorem. Let T / 4 be any real number greater than I and p ( t ) the T-periodic function defined by p(t) = 4T-'t if It( < T / 4 and p(t) = - 4 T - l t 2 if
+
In Sections 4 and 5, Fourier-transforms are defined by fix) =
e-lx'f(t)
dt to be
TI4 < t < 3T/4;p is the well-known sawtooth function of physics. Let q(t) = p ( t - T / 4 ) ;the function q is positive-definite and has an absolutely convergent Fourier series: q(t) = c, cos (2knt/T),ck 2 0, ck = 1. Hencep(t) = dk cxp (2knit/T),where Id,l = 1. I f o i s thediscrete measure dk6 ( x 2 k n T - l ) where 6 is the Dirac measure, the11b = p . Letg, be anycomplexvalucd boul:dedcontinuous functionon theline whose Fourier-transform S = is carried by [-I, I]. The Fourier-transform of y' is i t s ( t ) = i ( T / 4 ) p ( t )S(t). Hence g,' = i (T/4)a * g, and llg,'Il < (TI41 llall IIg,llm = (T/4) IIyllm. Since this inequality is true for each T / 4 > I, Bernstein's inequality is proved.
1,"
z?: +
x+:
x:
6
5. Other inequalities of Bernstein type THEOREM V: Let (Ak)- m < ,< + be an increasing sequence of real numbers such that lim (Aj+ - S ) / k = 1 ID uniformly over j E Z . For each
,
0 < A,,, < ( k + 1) L if k 2 0 and k L < A,,, < 0 if k < 0. Cantor's diagonal process yields an infinite increasing sequence m ( j ) of integers such that + A; for each k 2 0. For all k E Z , 0 < 6 < Am,,+ < L ; hence 0 < 6 < A;+, - A; < L and the set A' of all A;, - lm,k \ - co < k < + co is a regular set of real numbers. The preceding proof shows that if A has a realization A,, k E Z , such that lim (Aj+, - Aj)/k = 1/D uniformly over j E Z , so does k-+m
A': for each E > 0, there is an integer k ( ~ such ) that k 2 k ( ~implies ) (D-I - E ) k < Aj+, - Aj < (D-I + E ) k. But we can find an integer I depending only on m such that for each k in Z , A,,, = A,,, - x , . Hence for each j in Z , each m 2 1 and each k 2 k ( ~ )we , have (D-I - E ) k < - Am,j < (D-I E ) k . Passing to the limit, we get (D-I - E ) k < A;+, - AS < (D-I + E ) k for j~ Z and k 2 k ( ~ )which , is the required inequality.
+
k-+m
I < nD, there is a positive constant C such that for any complex valued bounded continuous function g, whose spectrum lies in [-I, I ] , we have ldAk)l. 191 < R
-m 0 such that each interval of length L contains at least one point ofA. A realization cf A is a presentation of A as an increasing sequence A,, - co < k < co, of real numbers. We have 6 < A,,, - A, < L and all other realizations of A are of the form ,uk= A,-,, ( - a < k < + a ) A . sequence A,, m 2 1 , of regular sets of real numbers tends to a regular set A if there is a realization A,,,, - co < k < $ co of eachA, and a realization A,, - co < k < co, ofA such that A,,, + A, for each k . We do not require uniformity of convergence with respect to k .
+
+
LEMMA 6: Let A be a regular set of real numbers and x,, m 2 1, a sequence of real numbers. There exist a regular setAf of real numbers and a subsequence xk of x , such thatA - xh tends ton' when m tends to infinity. Proof: Let A,,, be a realization of A - x , such that, A,,, 2 0 for k 2 0 and A,,, < Ofork < 0. We haveil,,, < L,A,,-, 2 -Landhence
E* of E with respect to the weak-star topology a (E*, E). For each real t the mapping from Q to the complex field defined by S + $(t) is continuous. If a sequence (S,),, of distributions carried by [-I, I ] exists such that sup 131 , = 1 while sup lgm(Ak)l< llm, let x , be a real number
,
R
-m 0, I f(x)l < Cw (1x1,) almost everywhere with respect to the Haar measure on Q,. We define Y(Q,) to be the vector space of all test functions g, : Q, + C which are compactly supported and locally constant. The Fourier-transform of such a g, also has these two properties. For any f E Lz(Q,), the Fourier transform S = ;Ff off, in the sense of distributions, is defined by
S(x) dx (b) for each t in A,, the mass ofL(S) concentrated in t is a(t) where a is defined as in Section 8.7; (c) the Fourier transforms of S and of L(S) coincide on Z,; (d) for a constant C > 0. (IL(S)II < CIISII, where the norms are computed in E, . In fact we put S = f E Lz, f = ( I f ) * p and L(S) = The terminology is that of Section 8.6 and the easy verifications of (a), (b), (c) and (d) are left to the reader. For each k 2 1 and each Radon measure p on Q,, let p k be the Radon measure p (pkx); if p is atomic and if the mass of p concentrated in x E Q, is a(x), p (pkx) is also atomic and the mass of concentrated ,ukin P - ~ X is a(x). Let I, be the characteristic function ~ f p - ~ Z , . A sequence L,, k 2 1, of linear atomizing operators can be found such that, for each S in Em,Lk(S) = S, is a nice sequence of approximations of S ; each S, is carried by the harmonious subset pkA, of Q, and this explains the role played by harmonious subsets of Q, in atomizing processes. If S E E,, S = where f E L,"(Q,). We define Skto be where
sf,
sf,
sf.
sfkk,
Then (5.5)-(5.7) imply that (a) S, is a measure carried by pkA,; (b) for each t in A,, the mass of L,(S) concentrated in pkt is a (pkt) jp.ct+zp, S(X)dx; (c) the Fourier-transform of S and of S, coincide on P-~Z,; (d) for a constant C independent of S and of k Z 1, llSkll < CllSll where the norms are computed in E, = OL,"; notice that pdkZPis an increasing sequence of open subgroups of Q, whose union is Q,.
9. Mean periodic functions Let Embe this Banach space of distributions; llSll is the norm of 3 in L3QP). A linear atomizing operator L can be constructed with the following properties: (a) for each distribution S in Em, L(S) is a measure carried by A, = A, n [O, 1 + E], where E > 0 isJixed from now on;
The theory of mean periodic functions provides a remarkable interpretation of coherent sets of frequencies: A is a coherent set of frequencies if and only if each mean periodic function whose spectrum lies in A is bounded or, equivalently, if and only if each mean periodic function whose spectrum lies in A is almost periodic. The exposition follows that of [47].
9.1. Engineers using Laplace transforms to solve linear differential equations of the type T * f = 0 where T is a derivation operator have noticed that this Laplace transform, originally defined in some right half plane, has a meromorphic extension F to the complex plane. Knowledge of the set (z,),, of poles of F and of the set P, {l/(z - 2,)) of principal parts of F at once gives the solution f as a finite sum Pk(t) exp z,t. Using the same idea, we study convolution equations of the type p * f = 0, where p is a given complex valued Radon measure carried by a compact interval [a, b] of the line and f is an unknown complex valued continuous function on the line. A solution of this equation is called a mean periodic function. The main theorem is the following: to get all solutions of p * f = 0 it is sufficient to consider all solutions f of the type P(t) exp At, where P is a polynomial and A a complex number, all finite sums of these particular solutions and all limits of such finite sums for the topology of uniform convergence on compact subsets of the line. Similar results can be found in 11601.
,
I,,
9.2. The space V(R)
From now on, a function will be a complex valued continuous function on the line, without any growth condition at infinity; V(R) is the linear space of all such functions. The topology of V(R) is that of uniform convergence on compact subsets of R. A measure will be a complex valued Radon measure carried by a compact subset of the line. The linear space of such measures will be denoted by ?JJlo(R).Then ?JJl,(R) is the dual space of Q(R). A mean periodic function f is defined by one of the two equivalent properties : (a) f belongs to Q(R) and the vector space spanned by all translates off is not dense in the whole of V(R); (b) f is a solution of a convolution equation p * f = 0, where p belongs to ?JJl,(R). From our point of view it is more exciting to start with such a convolution equation and to regard f as an unknown function.
9.3. The Laplace transform off Following the engineers, we solve p * f = 0 by taking the Laplace transform off. We shall prove that this Laplace transform has a meromorphic extension F to the complex plane. The determination of the poles of F and of the principal parts of F at these poles will give the approximation Pk(t) exp zkt. off by sums In defining the Laplace transform off, the first idea is to consider the integral J," f(t) exp -zt dt. This is not satisfactory since a mean periodic function can have a very rapid growth at infinity (Section 9.14). Furthermore this integral does not furnish the meromorphic extension. Our definition o f F is more unnatural: let f +(t) = f(t) if t 3 0, f += 0 if t < 0, f -(t) = f(t) if t < 0 and f -(t) = 0 if t 0. Consider the element g of V(R) defined by g = f * ,u = -f - * p. The first definition of g shows that g is supported by [a, +a[and the second one that g is supported by ] - co,b] when f is supported by [a, b]. Hence we can define the Laplace transform of g, G(z) = j,b g(t) exp (-zt) dt; G is an entire function of exponential type. Let M(z) = S,b exp (-zt) dp (t).
XkyO
+
DEFINITION 4: The Laplace transform of the mean periodic function f is the meromorphic function F(z) = G(z)/M(z). To prove the consistency of this definition, we show that F is independent of the choice of p such that f * ,u = 0. Iff * p' = 0, p' E ?JJlo(R),we define g' = f * p' = -f- * p' and we have g' * p = g * p'; hence G'(z) M(z) = G(z) M1(z)and F(z) = F1(z). Iff has a Laplace transform in the ordinary sense in g e z > x,, this f ) equals F on 9Zez > x, . Laplace transform 9( In fact +
G(z)
=
S:
=
J a exp
g(t) exp - tz dt
-zsdp(s)
=
S.
g(t) exp - tz dt
J o ft(u)exp
-zudu
9.4. Our program is to determine f by means of the set of poles of F. We need the following theorem:
X: If the Laplace transform F of the mean periodic function f THEOREM is an entire function in the complex plane, then f = F = 0. Before proving this theorem, we note the following: an entire function can be the Laplace transform of a complex valued continuous function on the line, but not of a mean periodic function (mean periodic functions are very special continuous functions on the line). 9.5. The proof of Theorem X depends on a series of easy lemmas and on a deeper proposition. LEMMA18 : Let f be a mean periodic function and o an element of n o @ ) . Then the Laplace transform of the mean periodic function f * o and the product of the Laplace transform o f f with jRexp -zt do ( t ) difer by an entire function. Proof: We have f+ * o = ( f * o)+ + r, where r is a compactly supported bounded function. Assume that f * p = 0 and put g = f + * p, h = ( f * o)+ + p. Then g * o = h r * p, which gives G(z)S(z) = H(z) + R(z) M(z), where R(z) = jRexp ( - z t ) r(t) dt is an entire function. The Laplace transform off is G(z)/M(z)and that off * o is G(z) S(z)/M(z) - R(z), which completes the proof.
+
LEMMA19: Let p be a complex valued Radon measure carried by [a,b] and A a complex number such that j : exp -It dp ( t ) = 0. Then there is a complex valued function m with bounded variation on [a, b] which is 0 outside [a, b] and such that, for each complex z f 1,
J
a
exp (- zt)m ( t ) dt = -J e x p - ~ t d p ( t ) . z-1 0
Proof: Replacing dp ( t ) by exp (-1t)dp (t),m(t) by exp (-1t)m ( t ) and z - 1 by Z, we can assume that 1 = 0. We simply take m(s) = y-, dp (t). From now on we shall write p~c,instead of m(t) dt. LEMMA 20: With the hypothesis of Lemma 19, let f be a mean periodic function, F the Lapluce transform o f f and1 a complex number which is not a pole of F. Then f * p = 0 implies f * p, = 0. Proof: As in the proof of Lemma 19, we may restrict our attention to the case 1 = 0. Then using the notation of Lemma 19,f + m has a distri-
butional derivative equal to f * p = 0. Hence the continuous function f * m is equal to a constant a. If a = 0, the lemma is proved; if a f 0 , the Laplace transform off * m is alz Lemma 18 shows that 0 is a pole of F, which by hypothesis is not the case. 3 : Let p be an N-times dzTerentiable function carried by PROPOSITION be the sequence of all zeros of the entire function [a, b],N 2 4. Let (Ak)kaO M(z) = j!: exp ( - z t ) p ( t ) dt. There exist an integer n 2 0 and a complex constant c such that
where p, is dejined as in Lemma 19 and where the series is uniformly convergent on [a,b]. Proof: The idea is to take Fourier transforms of the two sides of (5.9) and to establish convergence in L1 (- co, + co) for these Fourier-transforms. Returning to the original functions, we obtain uniform convergence. First of all, let us write down the Laplace transform of each side of (5.9). We obtain
But (5.10) is a consequence of Hadamard's factorization of entire functions of exponential type: ~ ( z=) C e c Z z n n ?(1 - ~ 1 1 ~ taking ) ; the logarithmic derivatives we get (5.10) with uniform convergence on compact subsets of C. In proving convergence in L1 (- co, + oo) of the restrictions to the imaginary axis, the following lemma will be useful: LEMMA 20 : Let
We have for a constant C and all real y, ly( 2 1 ,
The sequence of zeros of an entire function of exponential type satisfies
1: 11k1-2< + oo ([16],th. 8.2, p. 328) and we have N > 4 (the value o f N is given by the regularity of p). Hence Lemma 20 gives Proposition 3. To prove Lemma 20 we have to consider three cases:
(a) lz - 1,1 > litk1/2implies IXk(z)l < 2/21 IM(z)l 14k(-2by the second form of X,; (b) llk1/2 2 lz - Akl > 1 implies IXk(z)l < 21zI2 IM(z)l lAkl-2, since bl 2 I&1/2; (c) if lz - A,[ < 1 and lil,l 3 2, then (M(z)/(z - ilk)( < sup 12'-11=
2
IM(z')/(z' - A,)], since M(z)/(z - A,) is an entire functicn. Hence (M(z')I, where we have here lz' - A,[ 3 1 SUP IX,(z)[ < 2 12'-zl= 2
and lz/Akl < 2 Iz/Ak12by the assumptions on z and 1,.But from a direct evaluation of M(z) = J,b exp -ztp (t) dt, we obtain IM(z)l < ~ e ~ ' l " l / ( z ( ~ , where z = x + iy. Substituting this estimate in the inequalities obtained in (a), (b), (c) we get (5.11) with perhaps another constant C. Note, finally, that if ]jlkl < 2 and lz - a,/ < 11,1/2 we get lzl < 3 and there is nothing to prove. 9.6. Proof of Theorem XI Assume that the Laplace transform off is an entire function and that f * p = 0; convolving this equation with a regular compactly supported function, it may be assumed that p is N-times continuously differentiable. Let (A,),,, be the sequence of zeros of M(z) = JR exp (-zt) p(t) dt. , 0 for all k 2 0. Lemma 20 shows that f + p ~ = By the uniform convergence of (5.9), we get f * (tp) = 0. Iterating this, we obtain f * (Pp) = 0 for all polynomials P. Let to be a real number such that p(to) # 0. Then the Dirac mass E,, concentrated in to is the limit of a sequence P,p in the weak-star topology a (%,(R), %(R)). Passing to the limit, we get f * = 0 and f = 0. 9.7. THEOREM XI: Let p be a complex valued Radon measure carried by a compact set of real numbers. The set of all solutions off * p = 0, where f is a complex valued continuousfunction on the line, is given by the following set of rules: (a) all solutions h of h * p = 0 of the type h(t) = P(t) exp At, where P is a polynomial and 1a complex number must be found; Pk(t)exp 1,t of solutionsfound in (a) must be (b) allfinite sums g = written, (c) all limits f of convergent sequencesgj, j 1 of solutionsfound in (b) must be taken where the topology is that of uniform convergence on compact subsets of the line.
I,,
Before giving the proof, a remark should be made. The degree m of P and the complex number 1concerned in (a) can be found as follows: 1is a zero of the entire function M(z) = jg exp -zt dp (t) and m is the multiplicity of this zero.
9.8. Now to the proof. Let E be the space of all functions g defined in (b); E and the closure E of Eare translation invariant; B i s determined by the following rule : the function h belongs to E if, for each v E %,(R) whose Laplace transform N(z) at least vanishes whenever the Laplace transform M(z) of p vanishes, the convolution h * v is identically 0. We have to prove that f belongs to $ and to this end we show that k = f * v is identically 0. This is the case if the Laplace transform K(z) of k is an entire function (Theorem X). By Lemma 18, K(z) - F(z) N(z) is an entire function. But, by definition (Section 9.3) of F(z), F(z) M(z) is an entire function. Since N(z) vanishes at least whenever M(z) vanishes, N(z)/M(z) is also entire. Hence K(z) is entire, k = 0 and f E l?.
9.9. The spectrum and the Fourier series of a mean periodic function Let f be a mean periodic function, F the Laplace transform off, (zj),> the sequence of poles of F, Pj (l/(z - z,)) the principal part of the expansion of F in a neighbourhood of zj and mj the multiplicity of the pole zj or the degree of P,. DEFINITION 4: With the above notation, putting zj = 2niAj, we define the spectrum off to be set of all pairs (4, mi), j > 1. DEFINITION 5: The Fourier series off is the formal series exp 2ni1,t. The argument used in Section 9.8 gives the following result:
Pj(t)
XII: For the topology of uniform convergence on compact THEOREM subsets of R, each mean periodic function is a limit of jinite sums fk(t) = PjSk(t)exp 2ni;lJt. The degree of P,,, equals the degree mj of P,, and the coefficients of Pi,, tend to those of Pj as k -+ + co. To prove this last remark, the following lemma will be needed:
zja
LEMMA 21 : Let A be the spectrum of a mean periodic function f and (fi),,, a sequence of mean periodic functions whose spectra lie in A and
9.11. It remains to study sets of zeros of entire functions of exponential type having a slow growth on the real axis. For a precise statement of Beurling-Malliavin's theorem, a new definition will be needed.
which converges to f uniformly on each compact set of real numbers. Then the Laplace transforms Fj offj tend to the Laplace transform F o f f in the following sense: on each compact subset K of complex numbers not containing a pole of F, Fj tends uniformly to F and if 1 is a pole of F of order m , then 1 is apole of F j of order < m and the coefficients of the Taylor expansion of F j in a neighbourhood of 1 tend to those of F. Proof: Let p be an element of '2R0(R) such that f * p = 0 and let M(z) = 1exp (- zt) dp (t). Let g = f * p, gj = f: * p, G(z) = j exp (- z t ) g(t) dt, Gj(z)= j exp ( - z t ) g j ( t ) dt. Thenfj * p = 0 since the spectrum of f, is contained in that off. The Laplace transform off is G(z)/M(z)and that off, is Gj(z)/M(z).The supports of gj are contained in a fixed interval [a, b] and gj tends to g uniformly on [a, b]. Hence Gj(z)-, G(z) uniformly on compact subsets of C, and lemma 21 is proved.
,
+
x,>
9.10. How to construct mean periodic functions? Let A be a sequence of pairs (A,, m k ) k a where A, E C and m k E N. Is it possible to construct a mean periodic function whose spectrum is A? Beurling and Malliavin [30]give a complete answer to this question. We shall state the results. The problem can be divided into two parts.
XIII: There is a mean periodic function whose spectrum is A if THEOREM and only if there is an entire non zero function A(z) defined in the complex plane and satisfying A(A) = 0 for each 1 E Aand IA(x)l < C ( 1 1 ~ 1 for ) ~ suitable C > 0 and N 0 ( x E R)
+
IA(z)l < C exp llzl for suitable C > 0 and I > 0 ( z E C ) . Proof: The condition is necessary. I f f is mean periodic there is a con?plex measure p carried by some compact interval such that p * f = 0. By definition of the spectrum A(z) = jR exp (-2nizt) dp ( t ) vanishes on A. We define the complex Fourier transform of a compactly supported distribution S on the line to be A(z) = JR exp (-2nizt) S ( t ) dt. The condition is sufficient. By Paley-Wiener's theorem, A is the complex Fourier-transform of a distribution S carried by [-I, I]. Convolving S with a suitable test function pl in 9 , we may assume that S is an element of 9 , not equal to 0, whose complex Fourier transform vanishes on A. It remains to construct f as a series 2: aktmkexp 2niAkt ( A = (A,, mk),>J, where a, > 0 tends to zero rapidly enough to ensure uniform convergence on compact subsets of R. Hence f * S = 0 and f is a mean periodic function.
6: The mean period associatedwith a setA ofpairs(A,, m,),, DEFINITION (A, E C , mk 2 0 ) is the infimum p(A) of lengths of intervals [a, b] such that there exists a non-zero complex Radon measure ,u carried by [a, b] whose complex Fourier-transform1 vanishes on A. Remark 1 : I f [a, b] is an interval whose length is less than p(A), each complex valued continuous function defined on [a, b] is the limit of a sePk(t)exp 2niAkt, where dOPk< mk quence of finite sums Proof: Apply the Hahn-Banach theorem. Remark 2 : If [a, b] is an interval whose length is greater than p(A), each mean periodic function f whose spectrum lies in A and which vanishes on [a, b] vanishes on the whole real line. Proof: Let p be a measure carried by [a', b'], b' - a' < b - a, such that the complex Fourier-transform of p vanishes on A. The problem is clearly translation invariant and so we may assume that a = 0 Let f be defined by f + ( t ) = f ( t ) if t >, 0 and f + ( t )= 0 if t < 0. Sincef vanishes on [0, b] and since p is carried by [a', b'] whose length is less than b, f * p = f * ,u = 0. Using the definition of the Laplace transform F off, we get F = 0. Hence f = 0 by Theorem I. As a corollary we have that any mean periodic function whose spectrum lies in A is fully determined by its values on a compact subset depending only on A. It is interesting to compute the length of such an interval of defin~tion.Beurling and Malliavin's theorem gives the exact value of this length in terms of the exterior density of A, which we shall now define.
I
+
+
9.1 2. A density due to Beurling and Malliavin Let Q be a set of open intervals on the line (the notation o E 0 denotes intervals of this set), and let T(Q) be the union of all squares T ( w ) of R 2 one of whose diagonals is on w E Q : if x is the middle point of cu and 21 the J X - xl < 1. length of o, T ( o ) is defined by I YI A set 0 of intervals is said to be negligable if ( 1 + x2 + y2)-I dx dy < + co. (Note that for any set of open intervals, T(Q) is an open set of squares and is a measurable set.)
+
The complex Fourier-transform of ,u is 9: C -t C defined by P(z) = J: exp - 2nizt d~
w.
For example, if there is an I > 0 such that the lengths of the o E Dare all less than I, 9 is negligable. Roughly speaking, 2 ! is negligable means that it is improbable that 9 contains arbitrarily large intervals. A complex valued (not necessarily bounded) Radon measure p on the real line is said to be regular and has the density a if, for each positive E, the set 9,of all open intervals o of length lo1 such that llwl-l jm dp - a1 2 E is a negligable set of intervals. We then write a = A (dp). To each setA of pairs (A,, mk)kzl of real numbers together with multiplicities, we attach a positive measure dNAdefined by m, 6 (x - A,), where 6(x) is the unit mass concentrated in 0 and we assume that k -+ A, is an injective map.
C,,,
7 : The exterior density ofA, d,(A) is inf ( A (dv)) over regular DEFINITION positive measures dv such that dv 2 dNA. THEOREM XIV: With the above notation, (1) The set A of pairs (A,, m,),, is the set of frequencies of a mean periodic function f(t) on the line if and only ifd,(A) isfinite; (2) The mean period associated with A is d,(A). The proof of this result is deep and difficult. Since this theorem will not be used in the sequel, the reader is referred to [30]. An easy corollary can be given. We recall that a set A of real numbers is called regular if there exists a d > 0 such that each interval of length d contains at most one point o f A and a D > 0 such that each interval of length D contains at least one point ofA.
,
4: For a regular A, the mean period p(A) of A lies in the COROLLARY interual [ l / D , 1Id]. Proof: Consider R as the union of all intervals [kd, (k + 1) d[, k E Z and construct a measure dv as follows: if [kd, (k + 1) d[ n A = {A}, unit mass is concentrated in L and if [kd, (k + 1) d[ does not intersect A, unit mass is concentrated in kd. Hence dv 2-dNAand dv is regular and has the density lld. We thus obtain p(A) < lld. To get p(A) 2 1ID, we merely apply Theorem 111. 9.13. Mean periodic functions whose spectra are contained in a given A
From now on, A is assumed to be uniformly discrete, i.e. there exists a d > 0 such that each interval of length d contains at most one point ofA. We assume also that A does not present multiplicity (all m, = 0).
is then the Frechet space of all mean periodic functions whose spectra lie in A ; the topology V is that of uniform convergence on compact sets of real numbers. It is remarkable that the vector space of all mean periodic functions is not complete in the topology V ; the vector space of all mean periodic functions whose spectra are contained in a given A for whichp(A) < co is complete for %? since there exists a fixed measure ,u with compact support such that ,u * f = 0 for all such$
+
9.14. Behaviour at infinity
All f in V Aare fully determined by their values on any interval I whose 1 1exceeds p(A). The fundamental problem is now the following: length 1 is there a method of computing or estimating the values off outside I in terms of the values taken by f on I ? The answer is given by the following theorem:
+
XV: Let o : R -, [l, co [ be a function andA a regular set of THEOREM real numbers: each interval of length d contains at most one point ofA and each interval of length D contains at least one point of A. The following assertions are equivalent: (a) each mean periodic function f whose spectrum lies in A is 0 (o(t)) a s It( tends to in$nity; (b) there exist a constant C and a compact set K of real numbers such that, for each mean periodic function f whose spectrum lies in A and for all real t, If @)I < Cw (t) sup If I. K
Proof: Let ~?8be the Banach space of all complex valued continuous functions which are 0 ( ~ ( t ) as ) It( -t co.The norm of such an f in is sup Iw-lf 1. Let I: V A-+ LA? be the injection defined by (a). The topology
+
R
of W A is defined by the fundamental sequence of neighbourhoods of 0, V , = { f E V A: sup If 1 < 2-,}. The closed graph theorem shows that I C-f,kl
is continuous. Hence the inverse image under I of the unit ball of 9 contains a V , for sufficiently large k and this gives (b). The converse implication is trivial. Example: Let 8 > 2 be a real number and A, the set of all finite sums &,Ok,E , = 0 or 1. If 8 is a Pisot number, all mean periodic functions whose spectra lie in A, are almost periodic functions as we shall prove later. If 8 is not a Pisot number, mean periodic functions whose spectra lie in A, are known as soon as they are known on an arbitrarily small
interval [0, I] but it is impossible to give estimates for these functions in terms of their values on a compact set of real numbers. To substantiate these two remarks, two observations can be made: (a) if 8 > 2 each interval [x - T, x + TI of real numbers contains o(T) elements ofA and the little o is uniform with respect to x ; (b) there exists1 a sequence P,(t) of finite trigonometric sums whose frequencies belong to A such that P,(O) = sup IPk(t)l = 1 while P, tends R
uniformly to 0 on each compact set of real numbers not containing 0. Assertion (b) in Theorem XV cannot be true, since for s 4 K, t -+ P, (t - s) converges uniformly to 0 on K and takes the value 1 at s. THEOREM XVI: Let A be a regular set of real numbers. The following three assertions are equivalent: (a) each mean periodic function whose spectrum lies in A is bounded; (b) each mean periodic function whose spectrum lies in A is almost periodic; (c) A is a coherent set of frequencies. Proof: (a) * (c) by Theorem XV. Since trigonometric sums are dense in VA for the topology of uniform convergence on compact sets of real numbers (Theorem XII) and since by (c) this topology is equivalent to that of uniform convergence on the whole real line, each mean periodic function in VA is almost periodic. Hence (c) * (b). It is obvious that tb) =, (a).
Then A is the Fourier transform of a positive function b which is 0 ( r 2 ) as Proof of: rT
Let I > 0 be such that inf b(t)
> B > 0 (if E 6 1/4d, such an lexists).
c-1,ll
Then Parseval's equality gives l/b~11;= I&eA aAA(x - A)I2 dx = ll All: I r o n lan12.Hence 2 jk1 IP(t)I2 dt 6 11~112 laAI2.Since the space of trigonometric sums is translation-invariant, we get, for each k E Z, e2 jI::-f::: IP(t)12 dt 6 ~ ~ A I I : laAI2and adding a sufficiently large number of these relations we get (5.12). Proof of:
xleA
xAeA
c1AeA Z Ian126
-T
IP(t)12 dt.
:::
Eke,
(5.13)
2
We start as before with I ~ ~$ P =I I j$i+ b2(t) IP(t)I2 dt = /[A11 la,12. Since b(t) is ~(t-') as it1 + +a, 11b~11226 jTTb2(t) 1p(t)l2dt T-lC, @ ' , lai12 (using 5.12). Since b(t) 6 1/2d, we get I?, IP(t)12dt > 4d2 (llA112 - T I C 3 ) lanI2.If Tis large enough, (5.13) is proved. Notice that the L2 theory is easy and has a metric form while the L" theory (equivalence between sup If 1 and sup If I) is deep and is connected x A E A
+
rneA C-T,TI
R
with the arithmetical structure ofA.
9.15. The L2 theory We say that a set A of real numbers is uniformly discrete if there exists a d > 0 such that the distance between any two distinct points inA exceeds d. THEOREM XVII : There exist a T > 0 and two constants C1 and C2 > 0 depending only on d > 0 such that, for each mean periodic function f whose spectrum is simple and lies in A, we have rT
znoA
where the Fourier series off is an,exp 2nih. By Theorem XII, we can restrict our attention to finite trigonometric sums P(t) = an exp 2niilt. Let A :R + [0, + co[ be defined by A(x) = sup (0, 1 - 21x]/d).
xi,,
See Lemma 6, Section 5 of Chapter IV.
10. Notes IfA is a set of real numbers and if E is a Lebesgue-measurable subset of the line, we write A 4 (E, 2) if there exists a constant C such that for each trigonometric sum P(t) = A aAexp 2niAt, lanlZ< C j, (PI2dt 6 C2 laAI2. For the case where E is an interval, this problem has been widely studied and the infimum of the lengths of those E for which A 4 ( ~2), can be computed [48]. If E is a finite union of intervals, the problem becomes quite difficult: A 4 (E, 2) implies (El > &Zi (A), where I 1 denotes Lebesgue measure and the definition of the upper density ofA and the proof are in 1541. The converse result is plainly false, even if A = Z. The L2 version of Theorem VII gave the Erst example for which IE ( > dens (A) implied 4 ( ~ 2). ,
C
CAE
182
SPECIAL SERIES (COMPLEX METHODS)
It would be interesting to give other examples for which the latter implication is true and where the arithmetical structure ofA is not so explicit. For lacunary sequenced, some partial results have been obtained. For each Sidon set A and each Lebesgue measurable E of positive measure, A 4 (E, 2) holds 1311, although this hypothesis on A is not necessary for this conclusion [32]. An n-dimensional version of Theorem XI is given in [60].
CHAPTER VI
SPECIAL T R I G O N O M E T R I C SERIES (GROUP-THEORETIC METHODS)
T l s chapter is divided into two parts. The first one is devoted to topological Sidon sets A of real numbers: each bounded function b :A -* C is the restriction toA of the Fourier-transform of a complex bounded Radon measure p on R. For almost periodic functions f whose spectra lie in A, there is a diffusion of regularity: iff is continuously differentiable on an arbitrary small interval, the derivative off exists on the whole line and is almost periodic. Sidon topological sets are stable coherent sets of frequencies; in general, they are not harmonious sets. In the second part, we show that if (A,),,, is an increasing sequence of real numbers such that A, # k for infinitely many k > 1 and A, - k -, 0 as k -+ + co,the setAofA,, k > 1, never has this property. A continuously differentiable almost periodic function f whose spectrum lies in A and whose derivative is not bounded can exist; A is not a coherent set of frequencies. The proof of the last result uses the determination of idempotent or semi-idempotent measures on a compact group.
A. Topological Sidon sets of real numbers 1. Definition and basic properties of topological Sidon sets DEFINITION 1 : A set A of real numbers is a topological Sidon set if each complex-valued bounded function dejined on A is the restriction to A of the Fourier transform of a complex valued bounded Radon measure p on R. THEOREM I: L e t A be a set of real numbers. Thefollowing two statements are equivalent: (a) A is a topological Sidon set; (b) there exist a compact set K of real numbers and a positive constant C such that, for all trigonometric sums P ( t ) = cl exp 2niAt, we have Iden Ic~l < C SUP IPI.
Ed,,
K
Proof: (a)
(b): The hypothesis on A yields a linear map T : l m ( A ) the closed graph theorem shows that T is continuous. Hence there is a constant C , > 0 such that for each bounded function b : A + C, we can find a measure p in M ( R ) satisfying the two conditions llpll < C1ll bll, and P(1) = b(1) for all il in A. Theorem VI of Chapter IV shows that A is a coherent set of frequencies. Hence there is a compact set K of real numbers and a positive constant C , such that, for each function b in B(A), we can find a measure p carried by K such that llpll Q C211bllBcA, and P(1) = b(1) for all 1 €A. Combining these two facts, we ficd a compact set K of real numbers and a positive constant C such that, for each bounded function b : A --+ C there is a measure p carried by K whose norm does not exceed C sup Ibl A and whose Fourier transform restricted to A is b. Now let P(t) be a trigonometric sum '&A c, exp 2ni1t. We define b : A -t C by setting b(1) = 1 if c, = 0 and b(1) = c,/lc,l otherwise, so that ]b(1)]= 1. Let p be a measure carried by K whose norm is less than C and whose Fourier-transform restricted to Ais b. Then jK P(t) dp ( t ) = cA P ( 4 = ~ Z E AIcA Hence xnsn IcAl G C SUP IPI.
2. Examples of topological Sidon sets
-, B(A) and
InEA
K
(b) (a) : Let b :A -+ C be any bounded function. Let d be the normed space of the restrictions to K of all trigonometric sums ElenC, exp 2 n i h with the norm sup IPI. Then b defines a continuous linear form L on K
8 by L(P) =
XA,A -c,$ (4;from (b) we get IL(P)l < CnEnlanl Ib(L)I < C ll b ll ,
sup (PI. By the metric form of the Hahn-Banach theorem, L can be exK
tended to a continuous linear form on the whole of C(K). Hence we get a complex valued Radon measure p such that llpll < Cllblla, and @(A) = b(A) for all 1 in A. Remark: A Sidon set A in R can be defined by the following property: there is a constant C such that for each trigonometric sum P(t) = CnoA cA exp 2ni1t we have Ic,l Q C sup I PI. A topological Sidon set A is then
In,
R
defined by the two conditions: A is a Sidon set and A is a coherent set of frequencies. We use the qualification 'topological' to distinguish 'topological Sidon sets7from Sidon sets, for example, the set of all 2-,, k 3 0, is a Sidon set but is not a topological Sidon set.
PROPOSITION 1 : Let (A,),, be an increasing sequence ofpositive real numbers such that A,+ 3 41, for all k B 1. Then the set A of all A,, k B 1 , is a topological Sidon set. The proof of this proposition depends on the following lemma:
,
,
LEMMA 1 : Let (b,), be any sequence of complex numbers of absolute value 1 and (A,),, an increasing sequence ofpositive real numbers such that A,,, > 41,. Then there is a real number s such that Is1 < 1/1, and lexp 2ni1,s - bkl < 1 for each k > 1. Proof: The inequality lexp 2ni1,s - b,l < 1 is satisfied when s belongs to the union of closed intervals I,,, of lengths 1/31, whose centers belong to a suitable coset of 2, '2. Since A,,, B 41, it is very easy to see that Hence each I,,, contains each I,,, contains a complete interval such a solution s. Proof of Proposition 1: Let P(t) be any trigonometric sum C, exp 2ni1,t. We define b, by bkck = lckl if ck # 0 and by 6, = 1 if c, = 0. Then P(s) = z,, where lzkl = 1 and lz, - 11 Q 1 which imply We z, > 3. Hence g e P(s) 2 &, I lckl and x k r 1 lckl Q 2 sup IP(t)l. (tlS2l-i It will be proved later that any finite unionA of topological Sidon sets is again a topological Sidon set whenever there is a positive d such that d, 1 EA,1' €A.Applying this result in our situation, we get inf 11 - 2'1 the following theorem:
Ckrl
xk,ll~kl
,
11: Let ol be a real number greater than 1 and let (A,),, be an THEOREM increasing sequence of positive real numbers such that, for all k 2 1 , A,,, > nil,. Then the set A of all A,, k 2 1 , is topological Sidon set. 3. Construction of remarkable measures associated with topological Sidon sets 3.1. If A is a topological Sidon set, every bounded function b: A -, C is the restriction to A of an element 6 of B(R) but this statement does not give any information about the values of 6 outside A. The following theorem shows that we can be far more precise about 8. THEOREM 111: L e t A be a topological Sidon set of real numbers. There is a constant C > 0 such thatfor eachpositive E , eachpositiveq, andevery bound-
ed function b :A + C, we can find a measure p with the followingproperties: Ilpll < C5-l SUP Ibl (6.1) A
$(A) I$(t)l
< E sup Ibl A
=
b(1) for all 1 EA
if the distance from t to A exceeds 7 .
(6.2) (6.3)
Notice that the norm ll,ull of ,u does not depend on q > 0. IfA is a set of integers, then A is a topological Sidon set if and only if c,. there is a constant C such that for every trigonometric sum P(t) = exp 2ni1t, we have &, Ic,l < C sup IPI. The proof of Theorem 111then
3.3. A 'random' space Q used in the proof of Theorem III Let { - 1 , 1 ) be the multiplicative group of two elements a n d 9 = { - 1,1) the product of k copies of (- 1, 1). The Haar measure d o on 9 is defined by the condition that each point of SZ is assigned the mass 2-,. The dual group of 9 is the group G of all sequences g = ( E ,, ...,E,) of 0 or 1, where addition is defined to be componentwise addition modulo 2. Each function f : SZ + C has a Fourier series
xneA
C0.11
gives the following result. For some constant C > 0 and for each positive E , there is a measure p carried by [O,2n] such that ll,ull < CE-l. $(A) = 1 if 1 EA, and /$(j)l < E if j E Z, j $A. If the norm of ,u were independent of E > 0, ,u would tend to a weak limit v as E tended to 0. Such a measure v is carried by [O,2n],$(A) = 1 when 1 EA and 9(j) = 0 for all other integers j. The existence of v is impossible by Theorem IX below.
Parseval's relation gives
3.2. In the proof, we can confide our attention to the case where b(1) = 1 for all 3L in A. Let ,ul be the measure obtained in this case. Each bounded function b :A + C is the restriction to A of the Fourier transform fl, of a measure p, whose norm does not exceed C, llbll m. Theorem I11 is proved if we put p = p1 * ,u2. Since A is countable, it can be presented as a sequence l . AS a first step, A is replaced by A, = ( 4 , ..., 1,) and a measure p, is constructed such that (1) ,uk is carried by a fixed compact set K, independent of k ; (2) llpkll < CIS; (3) @,(A) = 1 for all 1 €Ak; (4) I$,(t)l < E if the distance from t to A, exceeds q. Let Q(K) be the Banach space of all complex valued continuous functions on K and let o (%R(K),Q(K))be the topology of pointwise convergence of measures tested on such continuous functions. We can find a subsequence of the p, which converges to a measure p satisfying conditions ( I ) , (2) and (3) of Theorem 111(with b = 1). ,ukwill now be constructed. Let ( K , , C,) be suitable for A: this means that for each finite trigonometric sum P(t) = FLEAc, exp 2ni1t we have &,A IcAl < C1 sup IPJ. Then (K, , C,) also has this property for A, and,
3.4. 'Random' measures p, used in the proof of Theorem III
K1
from now on, we write p instead of pk and A instead of A,.
Since (Kl , C,) is suitable forA,, we can find a measure v, for each o in 9 such that IIvmII < C 1 ,v, is carried by K and $,(Aj) = oj far 1 < j < k. Let ,urn be defined by p, = j, v,,. * v,. dw'. This convolution is, in fact, a finite sum. Hence p, is a measure; Ilp,ll < c:, PO (Aj) r-1 = j, ojoj oj do' = wj JQ do' = o+ and, for all real t,
where sj
=
0 or 1 and (el.
1....
Ic ( t ;E ,
, ..., &,)I
< C: .
&k)
This last inequality shows that the Fourier coefficients of the 'random' ,, are more regular than the Fourier coefficients of the 'random' measure u measures v,; the regularity is required with respect to cu E Q. To prove this last inequality, note that
By Parseval's formula =
d2 ( t ;E,,
...,E,)
(el.
C
....ek)
Id ( t ;E , , ...,&, ) I 2
which proves (6.6).
< C12 . But c ( t ;e l , ...,E,)
3.5. Riesz products R,
We now define 'random' Riesz products R,: R -+ C by R,(t) = - 1 - i + n 2 1 + smj cos Ajt) i (1 + EWJ sin Ajt). The following statements are easily seen to be true. (a) R,= -1 - i + A,+ iB,, where A,>,O, B,>O; hence JR,I d A, B, 2; (b) Sf2 IR&)l dm < 4; OJ exp iAjt s2E r r (sl , ..., sk, t) mE: (c) R,(t) = E w?, where r (sl, ,..,E ~t), is a product of two or more cosines added to i times a product of two or more sines and where the dash means that the last sum + sk 3 2. is extended over all sequences of 0's and 1's such that s1 + The Fourier-transform1 of R,(t) (with respect to t) is the measure
+
+
nt
+
The Fourier transform of R, f is given by k
(R, f ) " = f
A
* SR,
mjf(t - Aj)
=s 1
+ s2
x'
m?
m?'gE1, ...,Ek(t),
+
2:
where g,,, ...,,, - p,,, ...,,, * f and hence llg,,, ...,ekll < 2; the last sum is extended over all sequences (s,, ..., E,) of O's and 1's such that s, + ... + Ek > 2. Then (6.8) and (6.9) give
where 6(x) is the unit mass concentrated in 0 and where p,,, ...,,, is an atomic measure whose norm does not exceed 2. 3.6. A function f
The function f is 1/2n on [-n, n] and 0 outside. Then Il f 11, = f(0) = 1, and if n is chosen large enough, ~J'(t)l < s whenever It1 3 17. With such a choice of n, we put [-n, n] = K2 . 3.7. A formula for p
Sf2
We define p to be p, * (R, f ) dm. This convolution is taken in !Dl(R) and , j denotes a finite sum. 3.8. Properties of p
p is carried by K,
11~11
jn lllu~ll l &f
+ K, + K2 = K which is independent of k 11 dm
9 ,
2
C: So llRwf 11 dm = C I ~ Q X RIR,(t) f(t)l dt. Reversing the order of integration and using the inequality Jf2 IRw(t)ld o < 4, we get llpll < 4 ~ 2 . The Fourier transform o f f is f ( x ) = Jf 2 e-lxY(t)dt.
If the distance from t to A, exceeds q, we use (6.1 1). Since ...,El' lc (t; and ~g&ls...~&k(t)~ G 2, we have Ib(t)l < 2c2 ~ . f (-~ El . . zk)l < 2 ~ 2 ~ ~ If t = A,, 1 < j < k, then ,A,(t) = 0,. Using (6.10) we obtain $(t) = s. Finally p has the following properties: llpll 6 4 c f , /?(A) = s if A e A k and I,d(t)l G 2c;s2 if d (t,Ak) > q ; p is carried by K = K, + Kl + K2 which is independent of k. Replacingp by s - l p and then E by s/CI h,we obtain the requiredp,. 4. The union of two topological Sidon sets From now on let A be a uniformly discrete set of real numbers. This means that there exists a d > 0 such that for all 1and 1' in A, A # A' implies 1 1- A'( 2 d.
190
SPECIAL SERIES (GROUP-THEORETIC METHODS)
THEOREM IV: Ifrl is the unionA, u A, of two topological Sidon sets, A is again a topological Sidon set. The proof depends on the following lemma:
2 :LetA be a set of real numbers, K a compact set of real numbers LEMMA and C a positive real number, and assume that, for any function b : A -P C, there is a complex valued Radon measure p such that (a) p is carried by K; (b) Ilpll < c ; (c) lg(1) - b(1)l < 4for all A in A. Then A is a topological Sidon set and for all trigonometric sums Ic,l < 2C sup ]PI. P(t) = '&,A c, exp 2 d t , we have
ILEA
K
Proof: Define b(31)to be c,/(c,l if c, # 0 and to be 1 if cA = 0.Then, for -c,,d (1) the corresponding p given by the lemma, JK P(t) dp ( t ) = = xnGnca (b@) - b ( 4 ) + ~ A ICAI.~ Hence A g e S,. P(t) dp ( t ) 2 5 ,A I CAI. But B e j, P(t) dp ( t ) < C sup IP(, which completes the proof.
I,,,
1
K
Two uniformly discrete sets of real numbers A and A' are said to be adjacent if there is a 1-1 mapping p : A-+A' such that p(31) - 1 + 0 as
+
1311 + oo. A set of real numbers A is a stable coherent set of frequencies if each adjacent A', including A itself, is also a coherent set of frequencies. THEOREM V : Stable coherent sets of frequencies are Sidon topological sets and conversely. For any closed set E of real numbers, let B(E) be the Banach algebra of all restrictions to E of the Fourier transforms of complex bounded Radon measures p on R ; llyllBcE)= inf {IIpII ;$ I E = Q)}. The proof is based on the following lemma: LEMMA 3: Let A be a topological Sidon set. Then there exist a positive 1, (21 1 can be found such that lim Aj+,,, - 3Lj = + CO. IjI-r+m
Remark: For m = 1 , this property may fail. The setA of all 2k, k > 0, and all 2k + 1 , k 2 0 is a topological Sidon set and for an infinite set ofj, lj+l - Aj = 1. The proof of Proposition 2 is based on the following lemma: LEMMA 6: Let A be a topological Sidon set. Assume that,for all trigonometric sums P(t) = &,A c, exp 2ni1t, lc,J < C sup I PI. Assume that m
I,,
R
is a positive integer, A a finite set of 2'" elements, B a finite set of m elements and that all the m2" sums a b, a E A, b E B are distinct. Finally assume that A contains A B. Then m < C2. Proof of Lemma 6: Let 9 be { - 1, I)", already used in Section 3. Elements w of 9 are sequences (w, , ..., a,) of + or - 1 . We can put 9 in a ( 1 - 1 ) correspondence with A ; let w -t a(o) be this correspondence. Consider the trigonometric sums P,(t) = w j exp 2nibjt, where B = {bl , . . .,b,) and P(t) = P,(t) exp 2nia(w)t. Then P(t) = - + B c(v) exp 2nivt, where c(v) = 1. Hence m2"
+
+
I,,,
=
Z ~ A +Ic(v)l < C SUP IPI. R B
which gives Lemma 6.
1';
+
We shall prove that sup IPI R -.
<J ~ P ,
R
K
Select 2" points among the 4, j~J', such that the corresponding Aj + Bare pairwise disjoint; let A be the set of these 3Lj and apply Lemma 6. We get m < 4C2 which ends the proof. COROLLARY 1: For each topological Sidon set A, there exists an integer m > 1 such that, for every T > 0, a partition A = Mo U k J A jcan be achieved with the following properties (a) M , is finite; (b) Card Aj < m ; (c) the diferent Aj + [- T , TI, j E J, are pairwise disjoint. Proof: We can clearly restrict our attention to A n [0, +a[.Making the corresponding partition for A n 1- co,01 and enlarging M , , we obtain the corollary. Assume that all 1 in A are positive. The integer m is defined by Proposition 2. There exists an integer k, such that k > k , implies A,+, - 1, > (2m + 1) T . Let J be the set of all j > k , such that Aj - 3Lj- 2 (2 + l / m ) T. For each j in J, let j' be the integer in J which immediately follows j ; then j' < j + m and putting Aj = {Aj, b+l, ...,Ay - we get the required partition of A. 7.2. A lemma due to Zygmund PROPOSITION 3 : Let a be apositive element of L1(R),n an integer 2 2 , and d a positive real number. Let E be the set of all (n, n ) matrices
H=[s(xj-~k)ll<j,k$n
196
SPECIAL SERIES (GROUP-THEORETIC METHODS)
for all increasing sequences ( x ~ j 0. The proof depends on the following lemma: LEMMA 7: Let f be un element of L2(R),and let j s n and d be as in Proposition 3. There is apositive E = E (f,n, d ) such that for every sequence ( c ~ j)G ~n of complex numbers c j f ( x - xj)l12 2 E Proof of Lemma 7 : Assume that a sequence c ~ ,1~ 0 a real number sufficiently large that jlrl>T I8(t)l dt < E' (in fact > 0). We apply Proposition 2 to get a decomposition of A, A = Mo Uj,z Aj, where CardAj < m and where the Aj $ [-T, TI are pairwise disjoint; Mo is a finite subset ofA. (2) A function d' used in the dejinition of p. Let d = dA be the infimum of the distances between distinct points ofA and let A' : R -+ [O, + a )be defined by A1(x) = k sup ( 0 , l - 31xI/d), where k > 0 is a constant such that j, A'(x) 8(x) dx = 1. (3) A measure p. Let c :A -+ C be a bounded function whose connection with b :A + C will be explained later. The measure L,A has the following three properties : (a) llpll < Cis-l sup A Ic(4I;
+.
+
(b) P(x) = c(2)A' ( X - 2) if I X - A( (c) I,d(x)I g E sup Ic(A)I otherwise.
< d/3, 2 EA;
A
(4) A function c :A + C and a measure dv. If Card A j = 1 and 2 EA j , b(2) = c(A).
198
SPECIAL SERIES (GROUP-THEORETIC METHODS)
If 2 < Card A j < m , we write Aj as an increasing sequence A,, .. .,A, (all of whose terms depend on j), 2 Q k < m, and for 1 < s < k, c (A,) = x, is defined to be the solution of the linear system
LEMMA 8: Let A be a coherent set of frequencies and K a compact set of K If 1 and sup If 1 are equivalent norms for all real numbers such that sup
where b : A+ C is the function that we attempt to interpolate in the Fourier transform of a measure carried by [ - I , I ] and f is the convolution B * A'. Proposition 3 shows that for a constant C2 depending only on I, d and m , we have (x,J < C2,(1 Q s < k)-we recall that Ib(A)l Q 1 for all A EA --, hence IbI Q C ~ / E ; /2(x) = c(A)A' ( X - A) if J X - ill Q d/3 for some A € A ; I/2(x)I Q C2&otherwise. We define dv ( t ) = a(t) dp (t). An estimate for l4(A) - b(A)l. If A € A j and Card Aj = 1, we write $(A) = (d * /2) (A) = jR,d (A - t ) B(t) dt as the sum of three terms I , , I, and I, ; I, = jdi,3,, ,d ( A - t ) B(t) dt, I2 = jIISCI t I /2 ( A - t ) B(t) dt and I3 = j l t l > T / 2 (1 - t ) B(t) dl. Then 1121Q C2c since 1/21 Q E on the domain of integration while jR IB(t)l dt = 1. On the other hand 1131Q C,E since I,dI Q C3/&and j I t I > T IB(t)l dt Q E'. The main term I, = b(A) j!:,3 Ar(t)B(t) dt = b(A). Hence 14(A) - b(A)l < E (C2 C,). If A € A j and 2 Q CardAj < m , we put A = Ajs; Q, is the union of Intervals [A, - d/3, A, + d/3], 1 Q r Q k, Q2 is the complement of Q, lying in [Ajs - T , Ajs TI andQ, is the complement of [Ajs - T , Ajs + TI. We write 4(A) = I , I, I,, where It = So, /2 ( A - t ) &(t)dt For the same reason as above, 1121< C2&and 1131Q C3& On each interval [Ajr - d/3, Ajr d/3], ,A is c(Ajr)A' (X - Aj,) Hence I , = cArf(Ajs - Aj-1 = b(Ajs) In all cases [$(A)- b(A)l Q C4&,where the constant C4 depends only on I and A. Proposition 4 is proved if C4&Q 4.
norms for all almost periodic functions f whose spectra lie in A u (0). Proof: If the lemma were false, we could find a sequence gk = a, of such almost periodic functions with the following properties:
R
almost periodic functions f whose frequencies belong to A. Assume that 0 does not belong toA. For eachpositive E let K, be the set of all real x whose distance from K does not exceed E. Then sup 1f 1 and sup If 1 are equivalent K.
+
+ + +
+
x:=,
7.4. Conclusion of the proof of Theorem VIII To prove Theorem VIII, it remains to add to A\M, the finite set M,. Since Theorem VIII is translation invariant onA, the following lemma can be applied:
(b) sup lgkl R
R
+fk
=
This second condition implies lakl Q 1. Taking a subsequence if necessary, it may be assumed that a, + a as k + + co. Then sup la + fki K.
+ 0 and the sequence of ( f k ) k p is a Cauchy sequence in %(K). Since
sup 1 fkl and sup I fkl are equivalent norms, fk converges, on the whole of R , K
R
to an almost periodic function f such that f = -a on K, and whose frequencies belong to A. For each h. Ihl Q E,f ( x + h) - f ( x ) = 0 on K. Hence f ( x + h) = f ( x ) on the whole of R , since the spectrum of f ( x h) - f ( x )is contained in A, and f is a constant. Since 0 # A there is only one possibility: f = 0. Hence a = 0 and sup Igkl + 0. We thus have a contradiction. R
+
B. Idempotent and semi-idempotent measures 8. Idempotent measures on 1.c.a. groups 8.1. Some definitions: idempotent measures, S(p), the coset ring Let Gbe an1.c.a. group,rthe dual group andM(G) the space of all bounded complex valued Radon measures p on G.
DEFINITION 2: A measure p in M(G) is said to be idempotent if the convolution product p * p equals p. This definition is equivalent to the following one: DEFINITION 3 : A measure p in M(G) is said to be idempotent ifits Fourier transform ,A takes only the values 0 or 1.
Let ,u be an idempotent measure; put S(p) = ( y E T ;P(y) = 1). It is equivalent to determine ,u or S(p). To give the characterization of S(,u) for idempotent measures ,u, another definition will be needed.
r
4: Let SZ be the coset-ring of defined in the following way: DEFINITION SZ is the smallest family of subsets of with the properties: (a) SZ contains all translates of open subgroups of by elements of i.e. open cosets of (b) SZ is stable with respect to finite unions, finite intersections and complements.
r
r;
r
r,
THEOREM IX: Idempotent complex valued Radon measures p on G are characterized by the property: the support S(p) of the Fourier transform ,ii of p belongs to the coset ring of r.
8.2. The trivial half of Theorem ZX PROPOSITION 5 :Each element E of Q is an S(p)for an idempotent measurep. Let A be an open subgroup of and H the annihilator ofA. Then r / A is discrete; hence H , the dual group of r / A , is compact. Let m H be the Haar measure of H normalized by m H ( H )= 1. Then m , can be regarded as an element of M(G) and has the following properties: (a) if y E A then GH(?)= 1 ; (b) if y &Athen &&) = 0. Hence A = S(mH). If E = A + y o , we define dp = xro(x)dmH ( x ) ; then E = S(p). To prove Proposition 5, it suffices to show that the family 9'of S ( p ) for idempotent p is stable with respect to finite unions, finite intersections and complements. If p and v are idempotent measures S ( p * v) = S ( p ) n S(v), S ( p v - p * v) = S(p) v S(v) and S (6 - p) = I'\S(p), where 6 is the unit mass concentrated in 0 and \ denotes complementation. Hence SZ' =, Q. The proof of the reverse inclusion SZ' c SZ is deeper.
r
+
8.3. Reduction to the compact case LEMMA 9: Let p and v be two distinct idempotent measures on G. Then Ilp - vJl 1. The same result holds if the Fourier transforms of p and v take only integral values. - 911, = 1. Proof: Jlp - vIJ 2
DEFINITION 5 : The support group of p is dejined to be the smallest closed subgroup of G on which p is concentrated. PROPOSITION 6: I f p is idempotent, the support group of p is compact. Proof: Let H be the support group of p. Consider p as a measure on H and let I" be the dual group of H. By definitionof H, p isnot concentrated on a proper closed subgroup of H , therefore for all y # 0 in x,,u # p. Hence IIxyp - pIJ 2 1. On the other hand IIxYp - pi] tends to 0 as y tends to zero in I". These two observations imply that T'is discrete. To see that I I x , , ~ - pll + 0 as y + 0, let C, for any positive E, be a . the neighbourcompact subset of G defined by j,,, dip] < ~ 1 4Consider hood V of 0 in I" defined by sup Ix,(x) - 11 < s/211pj1. Then, for any y
r',
K
SK
in V , llxvp - pll < 2 JG\K dlpl + ~I211pIIdlpl < E. Assume now that Theorem I X has been proved for any compact group H. Let p be an idempotent measure on G and H the support group of p ; let A be the annihilator of H a n d set S H ( p )= { y E T/A;@(y) = I), where p is regarded as a measure on H. If n : + r / A is the canonical homomorphism whose kernel is A, S ( p ) = n - I (S,(p)). If SH(p)belongs to the coset ring of r/A,then S(p) belongs to the coset ring of F.
r
8.4. 'Translations lemmas' We arrive now at the main part of the proof. The main tool in the proof (Theorem X below) is a refinement of the translation lemma of Helson ([15],p. 66, lemma 3.5.1). In all of what follows it will be assumed that G is metrizable which implies that the dual group is countable. On M(G) the weak-star topology o (M(G), C(G))is defined by the duality with the space of all complex-valued continuous functions on G. The unit ball Ilpll < 1 of M(G) is a metrizable compact convex set for the weak star topology. The trivial modifications necessary when G is not metrizable are left to the reader: all sequences should be replaced by nets. The following result can be expressed in two equivalent ways; as usual B ( r ) denotes the Banach algebra of Fourier transforms of all complex Radon measures p on G.
r
THEOREM X :Let p be a complex Radon measure on G, (y&*
a sequence
of elements of the dual group T, y,(x) the value at x of the character y, : G + T and dpk(x) the complex Radon measure y,(x) dp (x). Let p = pa p, be the Lebesgue decomposition of p : p a = f dx, for some f E L1(G)andp, is singular with respect to the Haar measure dx of G. Assume that p, tends to a complex Radon measure v in the weak-star topology. Then either lim llv - pkll = 0 ( k -t + oo) or the weak limit v has the following two properties: (i) lvl < Ips] and therefore v is singular with respect to dx; (ii) llvll < Ilpll (note that the inequality is strict). An equivalent way of stating this result will be given. Let v be an element of B(r) and (yk)k2 a sequence of elements of Assume that for each y in r,g, ( y - y,) tends to a limit y(y). Then either lim Ily(y) k++w - y~ ( y - yk)IIBcr)= 0 or y is the Fourier transform of a singular measure and I Y I I B c ~ , < I I v ~ I B V ) . Before proving Theorem X , a corollary must be given.
+
r.
COROLLARY 1 : I f the measure p and the sequence (yk)kal satisfy the i i of p takes only hypothesis of theorem X , and if the Fourier transform , integral values, then either ykp = v for sufjiciently large k or lvl < lpsl and IIvII < Ilpll. Proof: Lemma 9 shows that if ykp = pk is a Cauchy sequence, then p, = v for sufficiently large k.
8.5. Proof of lvl < Ipsl: If ( Y , ) ~ , belongs ~ to a finite subset of T,(p,) belongs to a finite subset of M(G) and for sufficiently large k , p, = v. In this case Theorem X is proved. Otherwise there is a subsequence of the y,, k 2 1, tending to infinity. The following lemma will be useful: LEMMA10: If y k , k > 1 , tends to injinity in r and f E L1(G), then f,(x) yk(x)f ( x ) tends to 0 in the weak star topology o (M(G),C(G)). Proof of Lemma 10: Since 11 fklll = 11 f 1 1 , it suffices to test the weak convergence on a total subset of C(G); for example we can test on all characters y(x), y ET.But in this case jG fk(x) Y(X) dx = f ( y - yk),which tends to 0 by the Riemann-Lebesgue lemma ([15], th. 1.2.4, p. 9). To prove IvJ < lp,I, it suffices to show that, for any g in C(G), g dvl < j~kldlpsl. But g dv = lim ~ G ~ ( x ) Y ~ ( x )=~ Plim( x~)G ~ ( x ) Y & x>
8.6. The inequality (ii) depends on the following two interesting lemmas: LEMMA11 :Let p be a complex valued Radon measure on a compact space G, 8 a Borel function on G such that 161 = 1 and 8p = Ipl, E a positive number in (0, 1) and f a complex valued continuous function on G such that l l f llm G 1. Then We (jGf dp) )P ( 1 - E ) llpll implies llf - 811, G Jg IlpII, where the L1 norm is taken in the space L1 (dJpl)and We denotes the real part. We already know that llpll = sup We jG f dp; Lemma 11 locates Ilfllm~l
precisely those f for which this supremum is approached. Now to the proof. If lzl < 1, 1 - Wez 2 11 - zj2/2.Hence W e j Gf dp plG~~(l =%'eJG f;dlpl)P(l-E) ljpll i m p l i e ~ 2 - ~ ~ ~ I l - ~ f 1 ~ d l-EeOf)dlpl < (Ipll - ( 1 - IIpII < ~llpll.Schwarz's inequalitygives ( J G 11 - Of I dlpl)' < llpll JG 11 - Of l 2 dlpl < 2~llp11'. Since 181 = 1 , the required inequality follows. LEMMA12 : With the hypothesis of Lemma 1 1 , let (gk),>l , be a sequence of Borel functions on G such that lgkl = 1. Assume that gkp tends to a limit v in the weak star topology o (M(G), C(G)). Then either llv - gkpll -t 0 as k -, + a or llvll < llpllClearly llvll < Ilpll. We shall prove that llvll = IIpII implies the strong convergence of g k p to v. For any positive E , let f be a continuous function on G such that 11 f < 1 and W e JG f dv )P ( 1 - E ) IlpII. The weak-star convergence implies that for k 2 k(&),We jG fgk dp )P ( 1 - 2 ~ IlpII. ) Lemma 11 gives ll fgk - 811 < 2 & Ipll ; hence Ilf - g k e l l < 2 & lipll and for k and k' k ( ~ ) ,ll(gk, - gk) 811, = IIgk, - gkII < 4 J E 11.fiII. The sequence gkp is thus a Cauchy sequence and v is its strong limit. Returning to Theorem X , we apply Lemma 12 to gk(x) = yI(x) to terminate the proof.
=
SG
ISG
k++m
x f ( x ) d x + lim jcg(x) yk(x) ~ k++m
lJGg dvl
< JG Ig 1 d ! ~ # l -
k++m
( x )= lim jcg(x) ~ k ( xd~~ ) (x). Hence
P S
k+ C m
8.7. Proof of Theorem ZX From now on, a larger class F(G) of measures p will be used: F(G) is the subset of M(G) defined by P(r)c Z. It will be proved that for each p in F(G) and each rational integer a, the set of all y in such that P(y) = a belongs to the coset ring of r . Sketch of the proof: We have to show that on a compact group G each complex valued Radon measure p with integral Fourier coefficients can be written
r
where mj is the Haar measure of a compact subgroup Gj of G normalized by jGjdmj = 1 and Pj is a trigonometric sum over Gj with integral coefficients. The first term dMl = Pl(x) dm,(x) of this sum will be determined, up to multiplication by a character, by a minimal property: among all weak limits of products y(x) dp (x), y ~ rM1, has the smallest norm. Putting ,u = M, + M i , it will be shown that llpll = llMl[l IIM;II, llMlll 2 1 and 11 M i 11 < llpll - 1. Repeating this descent argument, we get (6.11).
+
r
For a p in F(G), d is the set of all measures y(x) dp (x) for all y in such that jGy(x) dp (x) # 0.All measures in &' have the same norm I(,u\(. Two cases may occur: (1) d is finite; then d = (7, dp, . ..,% dp). Let Go be the support group of p. Consider the Fourier coefficients of p with respect to the dual group of Go: all but n of them vanish. Hence dp (x) = x:pjyj (x) dm, (x), where m, is the Haar measure of Go normalized by mo(G,) = 1 and where the p j are rational integers. Theorem IX is therefore proved in this case. (2) at is infinite. Let 2 be the weak star closure of &' : 2 is compact and metrizable. Since jGdv is a rational integer different from 0 for all v E d , JjG dvl > 1 for all v E 2;hence 2 does not contain 0. The corollary of Theorem X shows that, if p E F(G), each v in 2\&' has the following two properties: llvll < Ilpll and v is singular with respect to the Haar measure of G.
8.9. An optimal measure v The norm decreases as we pass to the weak limits; hence there is a v in d whose norm is minimal. Let y be an element of r a n d ydthe set of all measures y(x) di2 (x) for all measures i2 in d. Since r is a group, y d is contained in 2. Let a be the set of all measures o = yv for which JG do # 0. Then 9? is finite, for otherwise the preceding argument would imply the existence of a o E 3 c d such that 0 < lloll < llvll and this is impossible by the optimal choice of v. We have 9 = -@,v, ..., T,,v), where y,, ..., y, ET.
8.10. DeJinition of the group GI and computation of v If ,L does not assume the value a on I', any translate of ,L also has this property. Since Z is discrete, B which is a pointwise limit of translates of $, does not assume the value a. Let p j = jGTj(x) dv (x), 1 < j < n. Then each p j is a rational integer and is a value assumed by ,L. Let G1 be the support group of v : regarding v as a measure carried by GI, all but a finite number of the Fourier coefficients of v vanish. Hence dv (x) = 2;pjyJ (x) dm, (x), where m, is the Haar measure of Gl normalized by dm1 (x) = 1-
IG*
8.11. Computation of the restriction M I of p to GI in terms of v
r
PROPOSITION 7: There exists a character y' in such that dM, ( x = y '(x) dv (x). The proof of Proposition 7 depends on the following lemma: LEMMA 13: Ifv is the weak limit of the sequence dp, (x) = y,(x) dp (x), then v is also the weak limit (in the topology o (M(G,), C(G,)) of the sequence yk(x) dMl(x). Proof of Lemma 13: We write p = M l -t M i , where M, is carried by GI and where (GI) = 0 TOprove Lemma 13, it suffices to show that ykM', = pCl; tends to 0 in the topology o (M(G), C(G)). For each positive E, there exists a continuous function f : G -t [0, 11 such that JG f l d ~ ; (< E and f = 1 on GI. Then for each g in C(G), IfG fg dp61 < sllgll,. Writing g = fg (1 - f ) g, we get, for sufficiently large k, (jG(1 - f ) gy, dp( < 8, since y, dp tends to v carried by GI on which (1 - f ) g = 0. Since (1 - f ) p, = (1 - f ) p;, we have ISG g dpkl < E (1 + JJgJlm ) for some sufficiently large k. Lemma 13 is now proved. Proof of Proposition 7: We apply Theorem X to the group GI and the sequence of measures y,(x) dM,(x), k 2 1. There are three cases to consider: (a) 11 ykdMl - vl) -t 0 as k + + co, which implies I] MI - 7,vll -+ 0 as k + + co. Since v EF(G), the Cauchy sequence (7,~)is stationary for sufficiently large k and M, = fkv. (b) the set of measuresy, dM, is infinite and does not converge in norm. But then its weak limit is singular with respect to the Haar measure of GI, and this is not the case here. (c) the set of measures ykdMl is finite. Then the weak limit v = ykdMl for sufficiently large k. In all cases dM,(x) = yf(x) dv(x).
+
We have proved that dMl ( x ) = z:pjyS ( x ) dm, ( x ) , where the pj are rational integers and dm, is the Haar measure of the group G I . I t may be observed that d M , is a weak limit of measures y(x) d p ( x ) for y E T , and that (1 M l (1 is minimal.
belongs to the coset-ring of r. Hence there is an idempotent measure o' such that o' p and Theorem XI will be proved. From now on, it will be assumed that p E F+(G).TWOcases may occur.
8.12. At this stage, we write M i = p - M I . Then llMill < 11p11 - 1. Everything that was done with p can be repeated with M i . After a finite number of steps, we get p = M I + + M,, where each M j has the same form as M I . The set of ally for which @ takes a given value a can be defined, starting with the sets E (b,j), where &fj takes the value b, by means of the operations of finite union, finite intersection and complement: write all decompositions a = b, + ... + b,, where bj E Z and lbjl < Ilpll; then the set E(a) of y for which @ takes the value a is the union, taken over all these decompositions, of the intersections E ( b o ,0) nE (b, , 1) n ... n E (b,, m). Now each E (b,j ) belongs to the coset ring of r and so, therefore, does E(a). Hence Theorem IX is proved.
9.1. The periodic case
9. Semi-idempotent measures Let r be a countable subgroup of R; we consider r a s a discrete abelian group. Let G be the compact dual group.
6: A complex radon measure p on G is said to be semiDEFINITION idempotent if for all t > 0 in T, we have @(t)= 0 or @(t)= 1.
-
Let r+ be the set of all y > 0 in r a n d p, :T+ -+ Z the restriction t o r + of the Fourier transform of p. The periodic case is, by definition, the case where a d > 0 can be foundinrsuch that for all y i n P , g, ( y + d ) = &). In this case, a measure o is given by the following lemma: lim [J(x)'Jn dp ( x ) LEMMA 14: With the preceding assumptions, n++m
do ( x ) exists; this limit is taken in the weak topology o (M(G), C(G')) and has the following two properties: o E F(G) and for each y in r + , 8 ( ~=) Lemma 14 clearly gives Theorem XI in the periodic case; d(x) is the complex conjugate of the value taken at x E G by the character d E Hom (G, T). Proof of Lemma 14: Put dp, ( x ) = [a(x)Ydp (x);then llpnll = IIpII and to show that p, has a weak-star limit o, it suffices to prove that @,(y) has a limit for each y in If y > 0, /&(y) = p, ( y nd) = y(y) for all n > 0. If y < 0, there is an integer m such that y + md > 0, as r is a subgroup of R. Hence for n > m , ,G,(y) = p, ( y nd) is constant. =
,w.
r.
+
+
9.2. The aperiodic case THEOREM XI: With the above notation, there exists, for any semi-idempotent measure p, an idempotent measure o such that 8(t) = @(t)for all t > 0 in T. As in the proof of Theorem IX, the hypothesis will be enlarged.
7 : Let p be a complex Radon measure on a compact group G DEFINITION whose dual r is a subgroup of R. W e write p E F+(G) i f @ ( y )E Z for all y > 0 in T. DEFINITION 8 : Let pl and p2 be two complex Radon measures on G. W e = for all positive y in r. write pl -- p2 It will be proved that for each p in F + (G),there is a o in F(G) such that o p. If p is semi-idempotent, the subset E of T,where 8 takes the value 1
-
If the hypothesis of Lemma 14is not satisfied, Ilyf(x)dp ( x ) - yV(x)dp (x)ll > 1 for distinct points y' and y" in r+. We define d to be the set of all measures dv ( x ) = Y(x)dp ( x ) for ally in r+such that JG dv ( x ) # 0; then JG dv ( x ) E Z. If no such y exists, it is plain that p 0 and Theorem XI is proved. Two other cases may occur: (a) d is$nite: Since for distinct y in T f , the corresponding measures dv are different, d is finite if there exist an integer n > 1, n distinct elements y , , ..,y, of r+and n non-zero rational integers p, , .. .,p, such that: if y = y j , 1 < j < n, @(y)= pj and if y # yj for all j, 1 < j < n, &(y) = 0. Then p -- C: pjTj ( x ) dx = o and Theorem XI is proved. (b) d is in jinite: Let d be the weak star closure of d.Then d is metrizable and compact and 0 6 d.
-
.
9.3. An optimal measure v and a number I The measure v is an element of 2 whose norm is minimal. There is a such that dp, ( x ) = Yk(x)dp (x) tends sequence (yk)kg of elements of to v in the weak-star topology. Passing to a subsequence if necessary it may be assumed that yk -t I 2 0 in the usual topology of R (I may be co).
r+
+
LEMMA15: Let g be the set of all measures de ( x ) = Y ( x )dv ( x )for all y > -I such that y E and jG de ( x ) f 0. Then the weak-star closure 3 of is contained in 2. is contained in 2. Writing dv ( x ) Proof: It suffices to show that = lim Yk(x)dp (x) and de, ( x ) = (7 7,) ( x ) dp ( x ) , we get d~
r
+
k++m
dek. Since y > - I and
lim y, = 1for the usual topology &-++a k++w of R, we have y y, > 0 for sufficiently large k and so the measure de, belongs to the set of all measures V f ( x )dp (x) for positive y' i n r . To prove that dpk E a2 it suffices to prove that jG dek f 0 if k is sufficienly large. But jG dek = 0 for an infinite set of k would imply jG de (x) = 0 , which is not true. Lemma 15 is therefore proved. =
lim
From now on, in the periodic case our optimal v will be replaced by the optimal yv, for a y ETsuch that JG yv, # 0. We can then write, in the periodic case, dv ( x ) = P,(x) dm, ( x ) . (2) The aperiodic case; G? is infinite: Since we are in the aperiodic case, Il y'v - y"vll > 1 if - I < y' < y". Let y,v, k 2 1 , be an infinite sequence of distinct measures in G? and let Q be the weak-star limit of an infinite subsequence of these ykv. Thene E d ,Q # 0 and, by Theorem X, Ilell< IlvII which contradicts the optimal choice of v. This case therefore does not occur. (3) The aperiodic case; G? isJinite: This means that, for all y > -I, 9(y) = 0 with the exception of n values of y, y j , for which pj = B(yj) # 0 (1 < j < n). We can then write dv ( x ) = pjyj (x)) d x dv' ( x ) ,where Bf(y) = 0 for ally > -1in.T.
(x:
+
+
9.4. Computation of v LEMMA 16: There exist a closed subgroup G , of G and a trigonometric sum P , with integral coefficients such that dv ( x ) = P,dm, ( x ) dv' ( x ) ,where m , is the Haar measure of G , and the Fourier transform of v' vanishes on I ' n (-1, + a ) . Three cases may occur in the proof: (1) Theperiodic case: There exists a d > 0 i n r such that for each y > - I, B ( y d ) = B(y). We define a measure v , by dv,(x) = lim [ a ( x ) ydv ( x ) ; then v , E F(G), gl(y) = 9(y) for all y > - I in and the product Yv, belongs to d for all y in I' such that jG7 ( x )dvl(x) # 0 , since y + nd > 0 for n sufficiently large. Since the norms decrease in the passage to weak limits llv, 11 < llvll; vl = 0 implies that 9(y) = 0 for all y > -1 (in this case $(y) = 0 for all y > 0). By the optimal choice of v, llvlll = llvll and the norm of each non-zero weak-star limit v, of yv,, y el', also has this property. By the proof of TheoremX, dv, ( x ) = P,(x) dm, ( x ) ,for otherwise dv, ( x ) would be a sum P,(x) dm, ( x ) + P2(x) dm, ( x ) + P,(x) dm, ( x ) and P,(x) dm, ( x ) would be a weak-star limit of products yv, such that 0 < IIPl dm1 ll < llv1 ll - 1.
+
+
r
+
9.5. Computation of p in terms of v From now on let G , be the support group of the measure dv (in the aperiodic case G , = G). LEMMA 17: With the preceding notation, i f P , is not identically 0 , dv is not singular with respect to the Haar measure dm, of G , . Proof: In the periodiccase, we have replaced v by v, = P , dm,, which is absolutely continuous with respect to the Haar measure of G , . In the aperiodic case we have B(y) = 0 for all y > - I in r with the exception of y = y j , 1 < j < n, for which B(yj) = p j . Then the following proposition will be applied ([15],th. 8.2.3, p. 200). PROPOSITION 8 : Let be a subgroup of R, G the compact dual group and o a singular complex measure on G such that 6(y) = 0 for a l l y > 0. Then jG do = 0. Proof: Let E be the closure, in the Hilbert space L 2 (do), of the vector space of all trigonometric sums P(x) = c o ayy(x). Assume for a moment that 1 belongs to E. This means that we can find a sequence P,(x) = a,(y) y(x), such that lirn j, 11 PnI2dlol = 0. Hence, by
r
zr
n-r
Schwarz's inequality,
1,
lirn n++m
+w
jG (1 + P,)
+
do = 0. But
jG P, do = 0
by
hypothesis; hence do = 0. To prove that 1 belongs to E, let p, be the orthogonal projection of 1 on E. Then 1 - p, is orthogonal to all elements y of E; applying this remark W ( X )= Y ( X )(l - ~ ( ~for 1 )y < 0 , we get jG y(x) I 1 - v(x)l2 (x) = 0. By complex conjugation this is also true for y > 0. Hence ( 1 - p,12
d(ol = cdx; since o is singular, c = 0 and g, = 1. Proposition 8 is thus proved. In our context, we can order the real numbers yj, (1 < j< n) in such a way that y, > yj for all other j. Then, if do (x) = y,(x) dv (x) is singular, B(y) = 0 for all y > 0 and jG do = p, # 0. This is impossible and so v is not singular with respect to dx. Lemma 17 is therefore proved.
9.6. Final stage of the computation of p Let M I be the restriction of p to G, and let p = M, + M i . If v is the weak-star limit of the sequence Vkp,then v, carried by G, ,is also the weakstar limit of the sequence V,M,, k > 1 (Lemma 13). It will be proved that VkM, = v for sufficiently large k ; hence by o, = lim y,v, (v, = P, dm,); we have Lemma 16, M, = lim y,v k+ + m k+ + m replaced y, by a subsequence if necessary, to ensure that the latter limit exists. Note that o, E F(G). Then M, E Ff(G), M i E Ff(G) and since llM',ll < llpll - 1, we reach the required conclusion ,u a, + ... + a, = o E F(G) after a finite number of steps. To show that v = ?MI, y > - 1, Theorem X will be applied. Two cases may occur. +a,we have IIM, - y,vll -+ 0 ; in the (1) If IljjkM1 - vll -+ 0 a s k periodic case v = v, = P,(x) dm, and y,v, k 2 1, cannot be a Cauchy sequence without being constant. Hence M, = ykvfor sufficiently large k. In the aperiodic case Ilyl'v - y'vll > 1 for y' # y", and the conclusion is the same. (2) If v = lim ykM, in the weak-star topology but not in the strong
-
Proof: The notation of the proof of Theorem XI will be used: a2 is the set of all measures X,(x) dp (x); if a2 is finite, A,,, - A, assumes a finite number of values and since A, - k tends to 0, A,+, - A, = 1 for sufficiently large k, which implies A, = k for sufficiently large k. If &is infinite, the optimal measure v is a weak limit of a subsequence of X,(x) dp (x). If x = A,, - A, for some fixedj and sufficiently large k in the subsequence, $(x) = 1. Otherwise $ = 0. Hence the support S(v) of 9 is contained in Z and v is idempotent. Let G, be the support-group ov v and let M, be the restriction of p to G, . Then dM, (x) = A,(x) dv (x) for infinitely many k. Hence S(M,), the support of M,,is contained in Z. Let M', = p - M I . Then S(M;) cAuZ. Repeating the preceding argument, we obtain a decomposition ,u M I + ... + M,,, of p, where the spectrum of each M j is contained in Z for
-
1 < j < m - 1 and the spectrum of M, is finite. Hence A, = k for sufficiently large k.
N
-)
,++a,
topology, v is singular with respect to the Haar measure of GI. We have seen that this is not the case. 9.7. A particular case be an increasing sequence of positive real num9 : Let PROPOSITION bers such that A, - k + 0 as k + co.L e t r be the subgroup of Rgenerated by the A,, k 2 1, and let G be the compact dual group. Assume that a complex Radon measure p on G has the followingproperties: $(A,) = 1for all k 2 1 and,ii(t) = 0 ift > 0 does not belong to the set d OfAk, k 2 1. Then A, = k for sufficiently large k.
+
10. Behaviour at infinity of special a-periodic trigonometric series THEOREM XI1 : Let (A,),, be an increasing sequence of positive real numLetA be the set of all A,, k 2 1 bers such that A, - k + 0 as k -+ +a. and V, the space of all mean periodic functions whose spectra are contained in A. If A, # k for an infinite number of k > 1, there is a mean-periodic function f whose spectrum lies in A which is not bounded. Proof: If all mean periodic functions in VAare bounded, A is a coherent set of frequencies. Hence a positive 1 can be found such that the Banach algebras B (A [- 1,1]) and B (A x [-I, I]) are isomorphic (see Chapter IV, Theorem X). For each E > 0, let A, be defined by A,(t) = sup (0, 1 - Itll~).Then A, belongs to A(R) and llA,llA(R) = 1. Applying TheoremX of Chapter IV, we find a constant C and, for each E in (0, I), a complex Radon measure p, on R, such that Ilp,ll < C and (A, + t) = A,(t) if k 2 1 andltl G I . Replacing 1 by a smaller number if necessary, it may be assumed that 1 < 3.Now assuming this, let T,(t) be the I-periodic function defined on [-5, +] by Tl(t) = 2AZl,,(t) - A1,,(t); then T,(t) = 1 on [-113, 1/31 and T(t) = 0 outside [-2113, 21/31. The Fourier coefficients of Ti are 0(k-2) and so TI is the Fourier-transform of a discrete measure o carried by Z.
+
P,
For k 2 k,, lAk - k( < 116. Choose E in (0,1/6); then v, = p, * o has the following properties (i) llv,ll < C2, where C2 is independent of E ; (ii) if t 2 Ako and It - 3Lk( d E, then P,(t) = A,(t); (iii) for all other t 2 Ako7 &(t) = 0. Let I' be the discrete group generated by A. Let j :I'-t R be the canonical injection and h: R + G the dual map. Then h is injective and h(R) is dense in G. When R is thought of as a dense subgroup of G, the bounded measures v, become complex Radon measures on G. In the weak-star topology (a(M(G), C(G)), a subsequence of the v, tends to a measure v with the following properties: (i) llvll < Cz, (ii) 5(t) = 0 for t 2 ilk,, t # A,, and (iii) $(Ak) = 1 if k 2 k,. Proposition 9 shows that ilk = k for sufficiently large k, and Theorem XI1 is thus proved. Letf be an unbounded mean periodic function whose spectrum lies in A. We write f(t) a kexp 2niAktwhere it may be assumed that all the Ak are different from zero. Then (ak12< + co and the primitive g off has the absolutely convergent series (ak/2niilk)exp 2ni3Lkt;hence g is almost periodic and continuously differentiable; the spectrum of g lies in A but the derivative of g is not bounded.
- xka
x,, xk2
11. Notes Helson sets are the compact versions of Sidon sets and are defined by A(E) = C(E). The family of Helson sets is stable with respect to finite unions [94]. Theorem IV is a remarkable discovery of Drury [40] and Theorem VIII was found by Mme Deschamps-Gondim [38]. A simple proof of P. J. Cohen's theorem (Theorem IX) was obtained by It8 and Amemiya [53] and the characterization of semi-idempotent measures can be found in [52].
CHAPTER VII
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
Let 0 > 2 be a Pisot number and E the compact set of real numbers of Cantor type constructed with the dissection ration 0-l. Then for each bounded continuous function pi :R -t C whose spectrum lies in E, there exists a sequence y,, k 2 1, of finite trigonometric sums whose frequencies belong to E such that pi, -t uniformly on each compact subset of the line and sup [pik[ d sup IyI. This is a strong form of spectral synthesis. R
R
In Sections 1-4 classical facts about spectral synthesis are recalled. In Sections 6 and 7 a fundamental theorem is proved, and in Section 8 a p-adic version of the preceding results is given.
1. Spectral synthesis and structure of closed ideals of a group algebra 1.1. The Banach algebra A(R)
Let A(R) be the Banach algebra of all continuous functions f : R -t C, which are the Fourier-transforms of functions in L1(R); A(R) is normed by Ilf I~A(R) = jzz I ~ ( ~ dl.) I Clearly 11 f (1, = sup If ( < 11 f [IA,) and (I f / I A cR) is translation-invariant. R
PROPOSITION 1: Each compactly supported continuously dzfferentiable function f : R -t C belongs to A(R). Proof: The hypothesis implies that f and f ' belong to LZ(R). Let F be the Fourier-transform o f f ; then the Fourier-transform o f f ' is - 2nixF and, by Parseval's relation, F and x F belong to L2. Schwarz7inequality dx)"' < + m and jlxlsl[F(x)~dx gives IF(x)l dx d Ji (j!, IF(" d (llxl, x-' dx)lIZ(jIxI, x21F12dx)lI2 d + m. Hence F belongs to
The following result shows how approximate units can be found in A(R) (A(R) does not have a unit).
214
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
215
THEOREM I: Let f and g be two functions in A(R). I f f ( 0 ) = 1, lim Ilg(x) - f(n-'x) g ( x ) l l ~ (=~ )0.
is the restriction to E of a function g of A(R), and the norm off in A(E) is the infimum, taken over all such extensions g, of llgllA(R).
If
PROPOSITION 2: I f f is a.function in A(R) vanishing at 0, the norm o f f in the restriction algebra A [-E, E] tends to 0 with E. Proof: Let or : R + [O, 11 be a function in A(R), which is 1 on [- 1. 11. Then, with E = n-l and the notation of Theorem I, Il f llAc-e, r l < 11 f ( x ) or (nx)llAcR)+ 0 as n -+ co. Since for each f in A(R), Il f llA[-&,&] decreases with E , the general case follows.
n++m
= 0,
lim
n++m
Il f (nx)A x ) I I A ~ R )
=
0.
Let F and G be two functions in L1(R)whose Fourier transforms are f and g and for each n 2 1, let Fn = nF (nx) and pn = n - lF (n- ' x ) . Then llFnlll = IIFlI . We have to prove that
,
In each case if we replace F by F' and G by G' such that ]IF - Ff 11 < E, IIG - G'II < E and if we put Fi = nFf (nx), we get /IFn- Fill < E and, for all n # 0, I(G - F, * G - G' FA * G'II < E ~ l l F l l ~ EIIGII + E~ while lipn* G * GfII1< ~ l l F / l ~EIIG,11 E ~ Hence . it suffices to prove the result for F and G belonging to a dense subspace of L1(R):we choose the continuous functions with compact support. If the supports of F and G are contained in [-T, TI, the support of G - F, * G is contained in [-2T, 2T] and, putting en = sup Ix-ylGn-1T IGb) - G ( Y ) ~ ,we get IG(x) - (F, * G) (41 = ~ j z ~G(x) : ~F,(y) ~ dy - jT1gInG(x- y) F,(y) dyl < ~,,llF,1 1 . Hence IIG - Fn * GII1 + 0 for these continuous compactly supported F and G and the proof of (7.2) is complete. Making a simple change of variables, we see that * GII, = /IF * G,IIl and returning to the case of continuous compactly supported F and G, we get, with the preceding notation,
+
+
+
+
+
+
3 :Let f be a function in A(R) and x a real number such that PROPOSITION f ( x ) # 0. There exists a function h in A(R) such that f ( x ) h(x) = 1 on a neighbourhood of x. Proof: By translation we reduce the problem to the case x = 0 and we may assume that f(0) = 1. If E < 1 , using the notation of Proposition 2, we get 111 - f llA[-c,d = 1101 - f II~c-e,el-* 0 with 8. For sufficiently small E, let g be a function in A(R) such that llgllAm < 4 and 1 - f = g on [ - E , E ] . Put h = x z g k ; this series isconvergent in A(R) and also for the uniform ncrms. Hence hf = 1 on [ - E , E]. PROPOSITION 4: Let K be a compact subset of R and f a function in A(K) which never vanishes on K. Then l/j'belongs to A(K). Proof: Let Y be the algebra of all complex valued functions all of whose derivatives, including the function itself, decay rapidly at infinity; Y is invariant under Fourier transform and is contained in A(R). For each x in K there exists a neighbourhood V(x)of x and a function g, in A(R) such that fg, = 1 on V(x).Standard arguments on partitions of unity show that there exist an integer n, n points x,, ..., x, of K and n regular functions or,, ...,a, in Y such that, writing V j for V ( x )and gj for g, if x = x j , each + is supported by V j and a , + ... + a, = 1 on K. We have: 1 = aj = ajgjf on K and llf = ajgj E A(R).
z: z:
x:
1.2. The ideals I(E) and J(E)
If Eis a closed set of real numbers, the restriction algebra A(E) is defined b y the following rule: a continuous functionf : E -,C belongs to A(E) iff
Among the closed ideals of A(R) ,a first type can be defined in the following way: let E be a closed set of real numbers and I(E) be the set of all f in A(R) which vanish on E ; I ( E ) is plainly a closed ideal of A(R). Let J,(E) be the set of allf in A(R) which vanish on a neighbourhood of E where the neighbourhood depends on f. Clearly J,(E) is an ideal of A(R) and so is the closure J(E) of J,(E) in A(R).
216
PISOT NUMBERS AND SPECTRAL SYNTHESIS
D E F ~ N ~1~: The O Nzero set of an ideal I of A ( R ) is the largest closed set on which all the,functions of I uanish. THEOREM I1 :Let I be a closed ideal of A(R) and E the zero set of I. Then J(E) c I c I(E). The second inclusion being obvious, it suffices to prove the first one. The proof depends on two lemmas: 1 : Let I be an ideal of A(R), E the zero set of I and K a compact LEMMA set of real numbers disjoint from E. Then there exists a function f in I which is identically 1 on K. Proof: For each x in K, there is a function f, in I such that f,(x) = 1, since otherwise x would belong to the zero set of I. Since I is an ideal, g, = f,f, belongs to I ; g, B 0 and there is a neighbourhood V ( x ) of x such that g, > -$ on V(x).Let V , , ..., Vnbe a covering of Kwith n of these V ( x )and let f be the corresponding sum gj where we write gj for g, and V jfor V ( x )if x = x j . Then f E I, and f ( x ) 2 for all x in K. Proposition 4 shows that there exists a function h in A(R) such that f h = 1 on K. Hence fh is the required element of I.
x:
PISOT NUMBERS AND SPECTRAL SYNTHESIS
217
2. Spectral synthesis and atomization of distributions 2.1. We begin with a technical lemma: 3: For any closed set E of real numbers, Y n Jo(E) is dense in LEMMA J(E>. By Lemma 2 it suffices to show that each function f i n A(R) whose support is a compact set K disjoint from E may be approached by a sequence f k of functions in Y. Since Y is dense in A(R), we can find a sequence g, in Y whose limit is$ Let ol be any function in 9which equals 1 on K and which is 0 on a neighbourhood of E. Then the sequence f k = ag,, k > 1 , tends to f as required. 3 : Let P M ( R ) be the space of all distributions S on R whose DEFINITION Fourier transforms 3 belong to Lm(R). We put llSllpM= 1115'11m and this norm is called the pseudo-measure norm. The duality between PM (R)and A(R) is dejined by
+m ( S
P)
=
(9, F )
S ( t ) ~ ( tdt,) F E L 1 ( ~ ) ,S E L'(R).
=
-m
2 : Let I be an ideal of A(R). The compactly supported functions LEMMA of I are dense in I. Proof: If ol is any compactly supported function A(R) such that a(0) = 1, then, for each f in I,fn = f ( x ) a ( n - l x ) belongs to I and tends to .f. Proof of Theorem I1 : Since I i s closed, it suffices to show that Jo(E) c I. Lemma 2 shows that we can further restrict our attention to compactly supported functions g in J,(E). Let K be the support of such a g ; K does not intersect E and by Lemma 1 there exists a function fin Iwhich equals 1 on K. Then g = g f belongs to I and the proof is complete. 2: A closed set E of real numbers is called a set of ~ynthesis DEFINITION if one of the following two equivalent statements is true: (a) each function f in A(R) which vanishes on E is the limit, in the norm of A(R) of a sequence f , of functions in A ( R ) which vanish on neighbourhoods Q k of E; (b) I(E) is the unique closed ideal of A(R) whose zero set is E. Theorem I1 shows that (a) e (b).
For any closed set E of real numbers, let PM ( E ) be the space of all S in PM ( R ) which are supported by E. LEMMA 4: For each S in PM ( E ) and each f in J(E), ( S ,f ) = 0. By continuity we may restrict our attention to f E Y n J,(E), and in this case ( S , f = 0 by the definition of the support of a distribution.
>
2.2. The weak-star topology o (PM(R),A(R)) In what follows o (PM ( R ) , A(R)) will denote the weak-star topology defined by the duality between A ( R ) and PM (R). A base of neighbourhoods of 0 for o (PM ( R ) , A(R)) is given by the sets V (P, E ) defined as follows: for each E > 0 and each finite subset F of A(R), V (F, E ) is the set of all S in PM (R) such that sup I(S, f )I < E. F
Since A(R) contains a contable dense subset, the restriction of the weakstar topology to the unit ball B of PM ( R ) is metrizable and B is compact for o (PM ( R ) , A(R)).
218
PISOT NUMBERS AND SPECTRAL SYNTHESIS
2.3. A description of P M (E) THEOREM I11: Let E be a closed set of real numbers. The following three properties of E are equivalent: (a) E is a set of synthesis; (b) for each S in P M (E) and each f in A(R) vanishing on E, ( S ,f ) = 0; (c) each distribution S carried by E whose Fourier transform is bounded is the weak star limit of a net (pa),,, of measures carried by finite subsets of E. Proof: To get the equivalence between (a) and (b), we consider the ideal J(E) as a closed linear subspace of Z(E). The Hahn-Banach theorem shows that Z(E) = J(E) if and only if (b) holds. The equivalence between (b) and (c) is also given by the Hahn-Banach theorem applied to P M (E) topologized by the weak-star topology o (PM (E), A(R)); let Mf(E) be the linear subspace of all measures carried by finite subsets of E. Then Mf(E) is dense in PM(E) if and only if (b) is true.
2.4. The dual of a restriction algebra DEFINITION 4: For a closed set E of real numbers, we define N(E) to be the subspace of all S in P M (E) such that (S, f ) = 0 for all f in Z(E).
5 : The dual space of the restriction algebra A(E) is N(E). PROPOSITION The proof is trivial.
PISOT NUMBERS AND SPECTRAL SYNTHESIS
(b) yk + g, uniformly on each compact subset of R, then y is bounded and continuous and the spectrum of y also lies in E. In fact let S be the Fourier-transform of g, and let F be any function in L1(R) whose Fouriertransform f vanishes on E. By Lebesgue's dominated convergence theorem
but plk(x) is a finite sum yk(-
X)
F(x) dx
xnGE a (k, A) exp 2niAx, and
=
A2
3. A strong form of spectral synthesis In certain cases that we shall now describe, the property of spectral synthesis gives a very concrete description of the space of bounded continuous functions pl : R -t C, whose spectra lie in a compact set of real numbers. 3.1. We begin with a very simple remark. If E is a closed set of real numbers and if a sequence yk, k 2 1, of trigonometric sums whose frequencies belong to E satisfies the two conditions
a (k,A) P(A) = 0
since f vanishes on E.
Hence (S, f ) = 0 and the spectrum of v, which by definition is the support of its Fourier-transform S, is contained in E.
3.2. Is it possible that, conversely, any bounded continuous g, : R -t C whose spectrum lies in E can be built up as the limit, in the sense of uniform convergence on compact sets, of a uniformly bounded sequence of finite trigonometric sums whose frequencies lie in E? THEOREM IV: Let E be a closed set of real numbers. The following three statements are equivalent: (a) for a constant C > 0 and each bounded continuousfunction g, : R -, C whose spectrum lies in E there exists a sequence v k ,k > 1, offinite trigonometric sums whose frequencies belong to E such that SUP1 ~ k G l c 1 ~ 1 R
1 : A closed set E of real numbers is a set of synthesis if and COROLLARY only if the dual of A(E) is PM(E).
219
R
(7.4) and vk(x) -t q(x) uniformly on each compact subset of Rr (b) for a constant C > 0 and each S in P M (E), there exists a sequence S k , k > 0, o f measures supported by finite subsets of E and such that, for each f in A(R)
(c) each S in P M (E) is the weak star limit of a sequence Skof measures carried by finite subsets of E. Moreover, the same constant C may be chosen in (a) and (b). 3.3. This theorem is surprising since (a) is very precise (for uncountable compact sets we cannot get better approximations) and (c) seems to be fairly general. However, the proof is very simple and uses only standard functional analysis.
220
PISOT NUMBERS AND SPECTRAL SYNTHESIS
(c) => (b) : Let B be the unit ball (ISIIPM < 1 of P M (E) and, for each
Q
> 0, let QBbe the homothetical ball llSllpM< Q.
The Banach-Steinhaus theorem shows that, if (S,, f ) -+ <S,f ) for < + a. each f in A(R), then sup llSkllPM
PISOT NUMBERS AND SPECTRAL SYNTHESIS
Ch. XVI, th. 4.19) the square mean
22 1
~ l,d12 dx is 0 and $ lim 1 1 2 STT T+
+m
cannot be almost periodic. Hence, if E is not countable, (a) cannot be improved to uniform convergence on the whole of R.
k> 1
Since A(R) contains a countable dense subset, the topology o (PM (R), A(R)) restricted to B is metrizable, and B is compact in this weak-star topology. Let B1 c B, be the set of all complex measures ,u carried by finite subsets of E whose pseudo-measure norms do not exceed 1;let C be the weakstar closure of B1: C i s clearly contained in B. By (c) and our first remark, B is the union of the increasing sequence D,, = B n nC, n ,) 1, of compact subsets of B. Since D, - D, c D,, ar,d B is compact, Baire's theorem shows that D, = B for n sufficiently large and (b) is proved. (b) (a) : For each real t , let p,, :E -+ C be defined by yt(x) = exp 2zitx. The mapping @ : R -+ A(E) defined by @(t) = p,, is continuous as (7.1) shows and, for each T > 0, the subset K of all p,, in A(E), for which It1 < T , is compact We apply the well-known result PROPOSITION 6: Let X be a Banach space and L,, k ,) 1, a sequence of continuous linearforms on X. Thefollowing three statements are equivalent: (i) for each x in X,L,(x) has a limit ay k -, + co; (ii) there is a constant C > 0 such that ((L,I ( d C and Lk(x)has a limit for all x in a total subset of X; (iii) L,(x) converges uniformly on compact subsets of X. (a) => (c): By Lebesgue's dominated convergence theorem, (7.4) implies yk(x)f(x) dx -+ jR y(x) f(x) dx for each f in L1(R), which is (c).
SR
3.4. We say that E is a set of strong synthesis if E possesses one of the three equivalent properties of Theorem IV. Varopoulos [94] gave an example of a set of synthesis which is not a set of strong synthesis, and Malliavin [61] gave an example of a closed set E of real numbers which is not a set of synthesis; we shall give another example in Chapter VIII. A countable compact set E is always a set of strong synthesis. Any bounded continuous p :R -+ C whose spectrum lies in E is, in this case, an almost periodic function and the approximation of p, by finite trigonometric sums whose frequencies lie in E holds in the norm of Lm(R). If E is not countable, E carries a diffuse bounded Radon measure ,LA, i.e. p has no discrete part; we put p,(x) = ,d(-x). By Wiener's theorem ([24],
4. Spectral synthesis and weighted approximation Using weight functions, a strong form of spectral synthesis can be defined where the approximation of the function by trigonometric sums holds in the strong sense.
4.1. L1(R, w), L" (R; w), C (R; w), Co (R; w) In what follows w : R -+ [l, + co[ is an even function which is increasing and concave on [0, + a [ , and we assume that w(0) = 1. For example o(x) = 1 + Ixla, 0 < a < 1, is such a function. We define L1 (R; w) to be the subspace of all f in L1(R) satisfying I f(x)l w(x) dx < + a ; this integral is equal to the norm o f f in L1 (R; w). If multiplication is defined to be convolution, L1(R; w) becomes a Banach algebra and A (R; o ) denotes the function algebra of all Fourier transforms of elements of L1(R; w). The dual space of L1 (R; w) is the Banach space of all complex valued Lebesgue-measurable functions a, on the line such that
l?:
Iy(x)l
< Cw ( x )
almost everywhere.
(7.5)
The norm of y in L a (R; w) is the infimum of all the C > 0 which can appear in the right hand side of (7.5). C (R; w) is the subspace of all continuous functions g, in Lm (R; w) and Co (R, w) is the subspace of all p in C (R; o ) such that p,(x)
=
o(w(x)) as 1x1 -+ +co.
(7.6)
4.2. Strong synthesis for weighted norms DEF~NITION 5: The spectrum of a function g, in Co ( R ;w) is the support of the Fourier transform S of p,. Note that g, is a tempered distribution. DEFINITION 6: A closed set E of real numbers has the property S(w) i f each continuous function p, : R -, C which satisjes (7.6) and whose spectrum lies in E is the limit, in the norm topology of Co (R ;w), of a sequence of trigonometric sums whose frequencies belong to E.
222
PISOT NUMBERS AND SPECTRAL SYNTHESIS
PISOT NUMBERS AND SPECTRAL SYNTHESIS
PROPOSITION 7 : If E has the property S(w)for all weights w, E is a set of synthesis. Proof: Let f be an element of L1(R); it is quite easy to construct a weight function w such that lim w(x) = + co and f E L1 (R; o). Let
bounded and if g, is a complex valued continuous function whose spectrum lies in E and which vanishes at infinity, (b) implies that sup R Ig,(x)
- y,(x)I 4 0 as k -+ + co; hence g, is almost periodic and this implies g, = 0. Since we have already proved this property of uniqueness in Chapter 111, we may assume that w(x) + + co with x Theorem V does not give a method of obtaining cpk in terms of g,. In the following statement cp, depends linearly on g, but we lose a little at (c).
Ixl-r+a,
p, be any bounded continuous function whose spectrum lies in E ; then p, E C,, (R; w), and S(w) gives a sequence
p,,(x)
a ( k , A) exp 2niAx
= AsE
such that
(7.7)
THEOREM VII: Let 8 be a Pisot number. There exist apositive constant C and a sequence L,, k 2 1, of linear mappings such that for any bounded continuous g, : R -+ C whose spectrum lies in E, (a) the frequencies of Lk(g,)belong to afinite subset F, of E; (b) I(Lk~) - dx)l Ce-klxl sup R 1~1;
Sincef E L1(R; w),(7.8)impliesjf: v,(-x) f(x) dx+ JT,"p,(-x) f(x)dx. Iffvanishes on E w e get j?: cp(-x) f(x) dx = 0 and E is a set of synthesis by (b) in Theorem 111.
(c) sup lLk~l R
5.1. The results Let 8 > 2 be a Pisot number a n d E the compact set of a l l s u m ~ ~ ~ ~ , O - ~ , = 0 or 1; E is a Cantor set constructed with the dissection ratio 8-'. The following theorems show that very nice forms of spectral synthesis hold for E.
E,
THEOREM V: Let 8 be a Pisot number and let v : R 4 C be any bounded continuous function whose spectrum lies in E. There exists a sequence vk, k 2 1, offinite trigonometric sums such that (a) for each k 2 1, the frequencies of p,, belong to E; (b) p,,(x) -+ y(x) uniformly on each compact subset of R; (c) l ~ k -* l (PI.
5.2. Sketch of the proofs Theorem VII will be an easy consequence of the atomizing properties of harmonious sets found in Chapter IV. By simple modifications jn the proof we shall get Theorem VI. To obtain Theorem V, we shall give in Section 7 a characterization of the restriction algebra A(E) in the style of Bochner's theorem ([15], p. 32, th. 1.9.1).
5.3. Proof of Theorem VII
R
EkaO
The proof will in fact show further, that the frequencies of cp, belong to the finite set of the 2, sums F ~ O -EJ~ ,= 0 or 1.
1:
THEOREM VI: Let 8 be a Pisot number, let co : R + [I, + a [ be an even unction which is increasing and concave on [0, + a[and let p, : R -+ C be a continuous function satisfying y(x) = o(w(x)) as 1x1 + + co. There exists a sequence p,, k 2 1, offinite trigonometric sums such that,for each k 2 1, (a) the frequencies of'p,, belong to E; Theorem VI simultaneously gives synthesis and uniqueness: if w is
c sup 1411. R
In fact, a little more will be proved: there is a finite subset F of E such that F, is thefinite set of allsums C: ~ ~ 0+- 8-,A7 j cj = 0 or 1, A E F; we note further that F, is contained in E.
5. Pisot numbers and spectral synthesis
R
223
1
Let A be the set of all finite sums Ekek,E , = 0 Or 1 and, for each k 2 1, let A, be the subset of the 2, sums 1:.@j, ej = 0 or 1. For each rational integer p, let zp be the isometry of LW(R)which carries p, into the function defined by x -+ p, (Opx). If the spectrum of 9 lies in E and if p = k 2 1, the spectrum of z,(v) lies in OkE= A, + E c A + E. The dilation x -+ Bkx is suggested by the right-hand side of (b). We recall Theorem XI of Chapter IV: let A be an harmonious set of real numbers and E a compact set of real numbers. There exist a finite subset F of E, a linear mapping L which maps bounded continuous functions g, : R -+ C whose spectra lie in A + Einto almost periodic functions
224
y
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
spectra lie in A + F and a positive constant C such that: for each il in A, the spectrum of y lies in il + F whenever the spectrum of p, lies in il + E.
= L(y) whose
For a function win the class Y of Schwartz, (7.11)can be improved to sup I Y IQ SUP I W Y I and (7.10) to I ~ ( x-)V ( X ) I < 1x1SUP I W Y I . (7.12) R
R
R
Starting with a y whose spectrum lies in E, the spectrum of t k y lies in A , + E c A + E. If 8 is a Pisot number, A is harmonious. We define Lk(y) to be ( z - , 0 L 0 z,) (91); (a) shows that the spectrum of ( L 0 7,) ( y ) is contained in A, F; hence the spectrum of L,(p,) is contained in O-,A, O-,F = F k . Statements (7.10) and (7.11) give the corresponding statements in Theorem VII.
+
+
5.4. Proof of Theorem V1 A first step in the direction of Theorem VI is the following Proposition 8. PROPOSITION 8: For each Pisot number 0 > 2 there exist a constant C > 0 and a sequence L,, k 2 1 , of linear mappings with the following properties: (a) for each continuous function p, in L m ( R ;o) whose spectrum lies in E, L,(y) is a trigonometric sum whose frequencies belong to E; (b) L,(f) +f uniformly on each compact set of real numbers; (c) I(Lk(f )>lo/ < Iflo/. R
R
We first show why Proposition 8 implies Theorem VI. If y ( x ) = o(o(x)), as 1x1 -+ + co,there exists a weight function s in the sense of Section 4.1 such that p,(x) = 0 ( a ( x ) )and D(x) = o(o(x)),as 1x1 -+ + a.By Proposition 8, I(Lkf)(x)l < CQ ( x ) for each real x ; this inequality and (b) together imply Theorem VI. 5.5. Proof of Proposition 8 Some properties of the weight function o will be needed. We put o ( x ) = 1 a(x); on [0, +a[, a is increasing, concave and a(0) = 0. It follows that a(x)/xis decreasing on [0, + co[ and
+
0 Q s Q 1 and t 2 0 imply a (st) 2 sor ( t ) s a l and t 2 0 imply a (st) Q sol ( t ) s 2 0 and t 2 0 imply a ( s t ) Q a(s) + a(t).
+
(7.13) (7.14) (7.15)
225
LEMMA 5 :Let E be a compact set of real numbers. There is a constant Cl such that,for each continuousfunction y : R -+ C in L" ( R ;o)whose spectrum lies in E
Proof of the lemma: By (7.14) we have 1 < o ( x ) < 1 + C l x ( ;hence if F E L1(R) and xF E L1(R),F belongs to L1 ( R ;o). Taking the Fourier transforms we see that iff E A(R) and f ', the derivative off, also belongs ) Ilf I ~ A ~ R ) + C l l f l l l ~ c ~Let ). to A@), then f A ( R ;o) and I l f l l A ( ~ ; r n6 a : R + [O, 11 be an indefinitely differentiable function which is 1 on a neighbourhood of E and 0 outside a compact subset of R. Put fx(t) = (exp 2nitx - 1) a(t). Using (7.1)it is easy to check that Il fxllAm < Clxl and 11 fillAcR)< Clxl. Let S be the Fourier-transform of p,; then - d o ) / = I<S,fx)l sup 1ylol l / f x l l ~ ( ~ ; c6u ) Cllxl sup Iflo(. R
R
Returning to the proof of Proposition 8, we have, if 1x1 Q 1, I p,(x) - p,(O)l < Cl 1x1 119 11. If 1x1 2 1 , a(x) and o ( x ) are equivalent; hence for all real x , Ip,(x) - y(0)l Q c2a (4 1ly11, IIvII = SUP I P ) I ~ I R --
If Proposition 8 is proved for y(x) - y(O), the result also follows for y(x). From now on, it will be assumed that, for some constant C , and
We define p, by yk(x) = p) (Okx)and a trigonometric sum P, by P, (Okx) = ( L [ y k ]()x ); L is defined by (7.9)-(7.12). The frequencies of P, belong to E as we have already observed in Section 5.3. If lxl 2 I ~ k ( ~ >=l IP (Okx)l < c2a ( O k x ) llyll Q c21x1 a ( e k ) lly11. If lxl Q l ~ l k ( ~ )= I Ip) (Okx)I Q CZa ( e k x )lldI < CZa ( O k ) IIvII. Since w in (7.12) decays rapidly at infinity, sup lwpkl Q C3a ( O k ) Ilp,\l R
and (7.12) gives Iyk(x) - (Ly,) (x)I < C31xl a(Ok) llplll, which, for 1x1 < 1 , does not exceed C3a (Okx)[IylI.Hence I(Lyk)(x)( < C5a(Okx)for 1x1 < 1. For all real x, (7.12) gives I(Lyk)( x ) ( < C4a (0,) liyII. A fortiori, l(Lyk)(x)l Q C,a (Bkx)IIyII for all real x. Finally IP,(x) - y(x)l - ~ and IP,(x)l < a(x) Ilyll, which proves Proposition 8. 1, ofjinite subsets of E such that the following implication is true f
E
C(E) and lim Il f
1, of linear maps such that, for each bounded continuous g, : Q , + C whose spectrum lies in E(8),
234
PISOT NUMBERS AND SPECTRAL SYNTHESIS
23 5
PISOT NUMBERS AND SPECTRAL SYNTHESIS
the frequencies of the trigonometric sum Lk(g7) are finite sums
Let P(X) E Z [XI be the minimal polynomial of 8: P(X) alXn-l + + an, where a, = pr = 181, (r 2 I).'
=
aoXn
+
m . 0
PROPOSITION 12: Regarded as a subgroup of TN, G is the group of all sequences o = (ok)k>lof complex numbers of absolute value 1 such that, for each k 2 1 and for real q~, ok= exp 2nicpk (7.41) and aoyk anvk+,= 0 (mod 1). (7.42)
+
+
With this representation of G, the duality between G a n d r is given by the in G pk8- in r and each o = following rule: for each y =
xk,
Proposition 11 is an easy consequence of Theorem XI of Chapter IV, and can be proved exactly as in the real case.
xv(o) = exp 2ni
pkcok. k> 1
8.4. The second state in the proof of Theorem IX is the following result, where E = E(8) is d e h e d as in Theorem IX.
r;
THEOREM X: Let f : E -t C be a continuous function and C a positive constant. The following two properties off are equivalent: (a) for any complex measure p carried by ajinite subset of E
(b) f is the restriction to E of theFourier-transform of afunction in L1(Q,) and Ilf IlAcE) < C. The implication (b) => (a) is obvious. The proof of (a) => (b) is very similar to that of Section 7. The idea is to construct a compact group on which& can be extended by continuity and to apply the Riesz representation theorem to describe the linear form P dp. +
I
LEMMA 10: Let m 2 1 be an integer. For each positive E and each sequence ( P ) ~ ksm ) ~ of numbers in A, satisfying (7.42), there exist an x in A, and m p-adic integers pk, 1 < k < m such that x8- + p k belongs to A, and
SE~
LEMMA 11 :Let I >, 1 be an integer and let w be an element of TNsatisfying (7.41) and (7.42) and such that ?, = 0 for I 1 < k < 1 f n. Then there is an x in p-"-'Z such that (7.45) x O - ~ = g l k ( m o d l ) for 1 < k < 1 n.
8.5. Definition of G
r
I,,
+
+ ..- + rk.
For each k 2 1, let rk be the group Z8-I Z8-k (without topology) and let be the inductive limit of the We regard r as a discrete group and denote its dual by G. Our discrete group r is a quotient of the direct sum of a countable set of copies of Z : to each finite sequence p = (p,),,, of rational integers corresponds pkrYkin Hence G is a subgroup of the compact product group TN of a countable set of copies of T.
Proof of Proposition 12: Let G' be the subgroup of TNdefined by (7.41) and (7.42). We describe an injective mapping i : G + G'. Each element x of G is a character on putting i(x) = (ok)k,l where ok= ~ ( 8 - we ~) get (7.42) from + ... + an8-k-n = 0 (k 2 I). Clearly i is continuous. = prZp. Let j* :r+ f' be the Let f' be the closure of F in Q,: identity mapping. Then j* is the dual of j: Ap/prZ+ G and i o j ( x ) = (exp 2 n i ~ € J - ~for ) ~ each , ~ x in A, (mod. prZ). To prove the surjectivity of i, from which it will follow that i is a homeomorphism, it suffices to show that I: A, -t G' defined by I(x) = (exp 2 n i ~ O - ~ ) ~has , a dense range. The following lemmas conclude the proof:
+ .
r.
I
i
. a = pr, r > 0, since P is minimal and all a&, E Ag.Let O, . .,6, be the other roots of P in LIP.Then 10lp > 1 and IOzlPG 1, .. ., 10nlpG 1 imply leip = 10 0, = lal/aolp 6 pr, If 101, < pr, the other symmetric relations between the roots of P give laJ/aolp< pr for 1 G j 6 n; p divides ao,.. ,a, and P is not minimal. Here It1 denotes the absolute value of the real number t E A,.
.
+ + +&Ip
PLSOT NUMBERS A N D SPECTRAL SYNTHESIS
PISOT NUMBERS A N D SPECTRAL SYNTHESIS
236
We begin with the proof of Lemma 11. Let GI be the group of all ~ E T ' + " defined by (7.42) such that p, = 0 for k 2 I + 1. Computing successively cp,, ql-, , ... we get pr possible choices for pl (mod 1) and for each of these choices, pr choices for 9,-, and so on. Hence Card GI = p". Let h : p-lr-rZp/p-rZp + G1 be defined by h(x) = (xO-~)~,,, where each term is taken modulo 1. Then, the kernel of h is equal to (0) and since Card (p-('+'" Z,/p-'Z,) = p", h is surjective. Since x is defined modulo p-'Z,, x may be chosen in p-(l 'Ir Z. To prove Lemma 10, it is convenient to consider first the case m = n. Let h : A, + Tn be the homomorphism defined by h(x) = (exp 2nixO-l, ...,exp 2nixO-"); then h(Ap) is dense in Tn. If this was not the case, a sequence p, , . ..,p, of rational integers could be found such that for each x in A,, x(p18-l + -.. + pn8-") G 0 (mod Z,). The only possibility is p18-l + + pn8-" = 0 ; as the degree of 8 is n, we getp, = = pn = 0. Hence no proper closed subgroup of Tn contains h(A,). If m 2 n + 1, m = n 1 say. We approximate (ql+l, ..., TI+,) by modulo ( x ~ e - ,~..-.,~x18-'-"). Then the difference (p, - x, 8-,), ,,, (Zp)m,is very close to an element of GI, which can be written as a sequence ( ~ ~ e - ~ ) ~ < modulo k < m , (Z,)", and x = x, + x, gives the requiredelement of A,. +
-
0
-
+
8.6. We return to the proof of Theorem X. For each j 2 1, let vj be the measure assigning the mass 3 to 0 and the mass 3 to 8-j, and for all j > 1 and k 2 0 let Pj,k = v j + l * " ' * vj+k. As in the real case, MjSkis the extension of ,4j,kto the whole of G and we have I+,
LEMMA 12: Iffor some fixed j, lim MI,,(@) f 0, then there is an x in k++m A, such that = ( ~ 8 ,)k2 - 1. Proof: If this is the case, lcos nqkl + 1 and 9, = p, + r,, where p, E Z and r, + 0. The congruences (7.42) give the corresponding ones for r, (when k > m). Let t , , ...,tnbe the complex roots of P(X) = 0. Then, + ..- + 2,~;~. Since rk + 0 as k + f a,we get for k > m, r, = 2, = ... = A n = Oandr, = O f o r k > m. Lemma 11 now gives the result. The last part of the proof is now a straightforward transcription of that given for the real case.
23 7
9. Notes
An excellent survey of the problem of synthesis for Banach algebras can be found in [14] (where sets of synthesis are called Wiener-sets). For 8 = 3, Theorem V was proved by Herz in [45]. The general form of Theorem V was conjectured by Salem.
ULTRA-THIN SYMMETRIC SETS
CHAPTER VIII
ULTRA-THIN S Y M M E T R I C SETS
We define a class of symmetric compact sets E of real numbers which are very similar to Cantor sets constructed with a dissection ratio whose inverse is a Pisot number. Such an E will be a set of uniqueness and of synthesis and all restriction algebras A(E) will be isomorphic; these E will be called ultra-thin symmetric sets. Other group algebras, like A(Dm), can be regarded as subalgebras of A(E) and, following Varopoulos and Katznelson [93, 111, this identification can be used to obtain a compact subset K of E which is not a set of synthesis.
1. Properties of ultra-thin symmetric sets We recall the definition of symmetric sets of real numbers (already given in Chapter 111, Section 2): we start with a compact interval [a, b] of real numbers and a sequence (t,),,, of real numbers belonging to the open interval (0, +); E is the ictersection n E, where E, is a union of kg1
2, intervals of length t , 5, ( b - a ) and is defined inductively by the following rule : to construct all the intervals of E,, each compact interval I of Ek-, is replaced by two compact intervals I' and I " of length tklI1 ;the left-hand end point of I' is that of I and the right-hand end point of I " is that of I; E, is [a, b]. E is called ultra-thin $27 5; < +co. As usual, the Banach algebra A(R) is defined by the rule: f E A(R) iff = Ilgll For each is the Fourier-transform of g E L1(R) and Il f compact set E of real numbers, A(E) is the Banach algebra of all complex valued continuous functions h defined on E which are the restrictions to E of functions f in A(R); IlhllAcE,= inf Il f llAcR, over suchf. If E is contained in the open interval (-n, n), and this is assumed in everything that follows, A(E) is also the Banach algebra of all restrictions to E of absolutely convergent Fourier seriesz?: ckeikXwhere C?: Ic,l < + co ; these two definitions of A(E) lead to equivalent norms. All ultra-thin symmetric sets are not affine copies of each other, nor
239
even locally affine copies. However, within the scope of harmonic analysis, all ultra-thin symmetric sets are the same as the following result shows: THEOREM I : All ultra- thin symmetric set^ E are sets of uniqueness and of synthesis and all the corresponding restriction algebras A(E) are isomorphic. The role played by the exponent 2 is crucial since, for each p > 2, we shall construct, following Salem, a sequence (tk)k,l of dissection ratios such that zy 5; < + co and such that the corresponding symmetric set F has the two properties: F is a set of multiplicity and for each ultra-thin symmetric set E, A(E) and A(F) are not isomorphic.
2. Functions whose spectra lie in ultra-thin symmetric sets 2.1. To get a more precise version of Theorem I, it is helpful to give another definition of ultra-thin symmetric sets. DEFINITION 1 : Let (tk)ka be a sequence of positive real numbers satisfying the two conditions:
for each k 2 1, tk > s , + ~= t k f l + tk+2 +
(Y)~
(8.1)
< +m.
and k, 1
The corresponding ultra-thin symmetric set E is the compact totally disconnected set E of all sums x = ZF cktk,E, = 0 or 1. 5k-l (1 - tk), a = 0, b = 1, we see that this new Putting t , = El definition is equivalent to the first one. DEFINITION 2: With the notation of De$nition 1, we denote by (a) F, thefinite set of the 2, sums cjtJ,E~ = 0 or 1; (b) E, the set of the 2, intervals whose origins belong to F, and each of whose lengths is s, + ,; (c) I , the interval [0,sk+J. We then have, Fk c E c F, I, for each k 2 1. We shall now describe the space of all complex valued bounded continuous functions y whose spectra lie in E.
x:
+
3 : For each k 2 1, we write p, as a perturbed trigonometric DEFINITION sum
ULTRA-THIN SYMMETRIC SETS
ULTRA-THIN SYMMETRIC SETS
240
+
where the spectrum of each g?, lies in I,, g?, = g,(Ei,...,Ek) if jl = t k t kand E , = 0 or 1, the second sum being extended over the 2, such sequences of 0's and 1's. For all real numbers x,xl , .. ., x , and with the same summation convention, we put
+
24 1
sake of simplicity, it will be assumed that
where we have v2 = 1 IT2. Now g,(t) becomes an almost periodic function with nine coefficients; the frequencies off are clvl E ~ V Zfor E , ,E~ = - 1,0 or 1. Better observations show that each of these nine coefficients off itself has 'secular perturbations' whose period T , is very large compared with T 2 and so on. If g, is assumed to have a 'nice' behaviour at infinity, for instance if g, is bounded, a mathematical model can be built to explain the 'projective sequence' of approximations of g,, to compute the corresponding error terms and to find the global behaviour of g,. Let E be the set of all infinite sums ekvk,E , = - 1 , O or 1. (We assume that the sequence T,, k 2 1 , of periods of 'secular perturbations' grows sufficiently rapidly to satisfy l / T k < + co.) Two cases have been studied in this book: (1) We have z y (v,, 1/vk)2< + co. Then this mysterious function g, is any complex-valued bounded continuous function on the line whose spectrum lies in E. For each k 3 1, we define the finite set to F, consist of the 3, sums &,vl + .- E,v,, E , = - 1,0 or 1. Then, for each k 2 1 , d t ) is identically zAEFkg,,(t)exp 2nijlt, where the spectrum of each g,, lies in [0,+,! ,I, where +,!, =vkfl V,+Z + -... We put g,, ( t o , t ) = C A E F k g,,&) exp 2niAt and -. we get Ig,(t) - g,, ( t o ,t)l < C (It - toll TT:~ sup IyI. Hence if L , = JT,T,+, and to = 0 , Ig,(t) - g,, (0, t)l
Taking the sup over Y', We get (8.6)
v ULTRA-THIN SYMMETRIC SETS
ULTRA-THIN SYMMETRIC SETS
Using exactly the same argument, the following result (used in Section 7) can be obtained.
LEMMA 1 : Let q > 1 , w > 0, 0 < I < llqw, let I be the interval [0, I ] and let h : R + T be the continuous homomorphism dejined by h(t) = exp 2niwt. There is a constant C depending only on q such that,for each pseudo-jiunctio~zS carried by I
244
2: For each ultra-thin symmetric set E of real numbers PROPOSITION there exists a constant C such that, for each sequence 81, ..., 8, of n real numbers, the function f : E -t C defined by
belongs to A(E) with a norm not exceeding C.
245
Roughly speaking, Lemma 1 asserts that the local isomorphism between R a c d T defines almost an isometry for pseudo-measure norms of distributions carried by a small interval. Proof of Lemma 1: Let S be a pseudo-function carried by I and let y(x) = S(-x). We have to prove that sup lyl < ( 1 + C12w2)SUP IqI; let R
mZ
4. Proof of Theorem 11: reduction to a problem on intervals
x , be a point where Iq(xo)l = sup 1471: such a point exists since q is
4.1. THEOREM 111: Let or, t l , ..., t, be an increasing sequence of n + 1 positive real numbers and let q > 1 be such that t1 > qol, t2 > q ( t , + a), ..., t , 2 q ( t , - l ... t , a); C will denote a constant dependingonly on q. Let A , be the set of the 2" sumsx: sktk,ck = 0 or 1 and let a,(x), ilELI,, be a sequence indexed by A, of complex valued bounded continuous functions whose spectra lie in [0,or]. We put
continuous and vafiishes at infinity Then. by Taylor's formula, q ( x ) = y(x,) ( x - x,) A 2-' ( x - x,)' B(x), where A = yt(x0), 1B1 < sup IqV(x)l.By Bernstein's inequality (Chapter V, Section 4 ) sup Iq"l
+
+ +
R
+
+
R
R
< 4n2I2sup lql = 4n212 Iq(x0)l. Sicce Iq~(x,)lis the maximum of all Iq(x)l, R
the minimum of ly(xo) + ( x - x,) A1 is obtained when x = x , and A/y(xo) is a pure imaginary (see Chapter I, Section 4.3, Lemma 7). We get (y(x)I 2 Iy(xo)l (I - 2n2121x - xoI2). There is an x G o Z such that Ix - x,l < w/2 which implies sup lql 2 (1 - n21202)/2)sup 1yl.l aZ
R
LEMMA 2 : Let q > 1 , 0 < I < ql < v, and let a,(x) and al(x) be two functions vanishing at infinity whose spectra lie in [0, I]. Then suplfl R
0, each integer m > 1 and each sequence p i , ... ,pn of rational integers satisfying I pjl < m , (8.18) lplvl + ... pnvnl 2 Dm-"+ l .
+
Roughly speaking, Theorem V asserts that the best estimate in (8.17) for n indeperldent frequencies holds if and only if these frequencies are strongly independent.
8
250
ULTRA-THIN SYMMETRIC SETS
COROLLARY: Let v, , ...,v, be n real numbers satisfying (8.18) and let (tk)k>lbe a sequence of positive real numbers tending to 0 sufficiently rapidly that (tk+l/tk)2/n< + co. Let En be the set of all sums for E , = O or v l , ..., or v,. Then En is a set of synthesis and a set of uniqueness and all the restriction algebras A@,) are isomorphic. The proof of the corollary is an easy transcription of the proof of Theorem I. Now to the proof of Theorem V.
z,,
I,,
6.2. Proof of (8.18) => (8.17). As in Lemma 1 , we can confine our attention to the case where the functions a,, .. ., a, vanish at infinity; we put
+ al(x)exp 2niv1x + + a,(x) exp 2nivnx f ( x ,y) = a,(x) + a,(x) exp 2nivl y + ... + a,(x) exp 2nivny
ULTRA-THIN SYMMETRIC SETS
25 1
for sufficiently small I and assume that 1x1 2 mT Let T = (the exact value of m 2 1 will be given in a moment); we get la (1x)l < 1 - am212'". Testing (8.17) on f(x) = a (Ix) P(x), where P(x) = co + C: cj exp 2nivix, we get sup [ sup
(P(x)l,( 1 - am212/")((P((,]2 ( 1 - Cl2In)I(P(I,.
1x14 m T
If am2 > C, the only possibility is
sup IP(x)l 2 ( 1 - Cl2In)IIPll m;
Ixl$mT
Theorem IX of Chapter I applies now give (8.18). The same argument shows that 12'" cannot be replaced in Theorem V by any number which is o(l2/")as 1 tends to 0.
f ( x ) = a,(x)
and M = sup I f(x, y)l. R2
Theorem IX of Chapter I shows that for each positive E and each interval I of real numbers of length C E - ( " - ~ )and I ~ for a fixed x,
Hence there exist two real numbers x, and yo such that Ix, - yO1 < c E - ( n - 1 ) / 2 and I f(x,, yo)l 2 ( 1 - E ) M . As in the proof of Lemma 1 , we have
7. From the group R to the group Dm
It is indeed remarkable that in the heart of the group algebra A(R), dwell other group algebras. For the sake of simplicity we shall confine our attention to the product group Dm of a countable number of copies of 2 / 2 2 and it will be proved that, for each ultra-thin symmetric set E, A(Dm)is a closed subalgebra of the quotient algebra A(E). 7.1. The group Dm and the group algebra A(Dm) Let D be the group with two elements 0 and 1 where addition is defined modulo 2 and let Dm be the product group D, of a countable set of
n
k>l
Multiplying f ( x o , yo) by a constant of absolute value 1 if necessary, we may assume that f(x,, yo) is positive. Then, putting A = a + ib, we get
+
hence ( x - x,) a < EM 2n212( X - x , ) ~M . By an optimal choice of x - x,, la1 < 2521 & M ; (8.19), with this estimate of a, gives 1 f(y,, yo)l 2 ( 1 - Cll2In)M when E = I2In, which is (8.17). Proof of (8.17) => (8.18). Let a : R -, C be a function in the class Y of Schwartz whose Fourier-transform is a positive indefinitely differentiable function supported by [0, 11. Assuming that a(0) = 1 , we get, for two positive constants a and j3, 0 < j3 < 1 , la(x)l < 1 - ax2 if 1x1 < 1 and la(x)l < j3 if 1x1 2 1.
copies of D. An element o of Dm is a sequence ( w , ) ~ of ~ elements of D; addition is componentwise addition modulo 2. The topology of Dm is that of componentwise convergence and the Bore1 field of Dm is generated by elementary sets E (a,, ...,a,) of the type ol= a,, .. .,con = a,. The Haar measure d o of Dm is defined by assigning to E ( a , , ..., a,) the measure 2-". For each n 2 1 , let xn : Dm + T be defined by xn(o)= (x, is a continuous character on Dw and each character x : Dm + T can be written x,,(o) = x;' X> for a suitable n 2 1 and a suitable sequence y = (a,, ..., a,) of 0's and 1's. The dual group of Dm is the direct sum of a countable set of copies of D. We denote by A(Dm)the Banach algebra of all complex valued continuous functions f : Dm + C whose Fourier series is absolutely convergent
-
r
252
ULTRA-THIN SYMMETRIC SETS
253
ULTRA-THIN SYMMETRIC SETS
7.2. A(Dm) as a subalgebra of A(E) Throughout this section, E is an ultra-thin symmetric set of real numbers. THEOREM VI : Let 17: E + Dm be the continuous surjective mapping dejinedby17(x) = o ifx = E?&,tk, E, = 0 or 1 and o = wk = 0 or 1, are connectedby wk = ezk-, E,, (mod2) for all k 2 1. Then for any complex valued continuousfunction f : Dm + C the following two assertions are equivalent: (a>f E 4 D m >; (b) f o 17E A(E). Before proving this theorem, some remarks should be made. Let H be the homeomorphism between E and Dmdefined by ~ ( x k I, &ktk)= (ck)kr 1 when ek = 0 or 1, k 2 1. Then H does not induce an isomorphism between A(E) and A(Dm). Using general theorems ([15], p. 78, th. 4.1.3) on homomorphisms of group algebras it can be proved that such an isomorphism between A(E) and A(Dm) cannot exist: A(Dm) is not isomorphic to the whole of A(E) but only to a closed subalgebra of A(E).
g = (ck)kal such that 82,-1 = E 2 k (k 2 1); G is clearly another copy of Dm and the Haar measure dm of G is a Radon measure on D ~ . For each g, E C(E), we define a continuous function F : Dm + C by F o H = g,, where H is the canonical homeomorphism1 between E and Dm, arid we put L(9) = y if, for each o = P(E),
+
7.3. Proof of (a) => (b) It suffices to show that there exists a constant C such that for each character xv: Dm + T, IJxY0 1711,(,, < C. If xY(o)= x:' ,:x g = xYo 17is defined on E by
.
g(x) = exp in (PI&, $.
.-- + /32n~Zn)when
m
x
=
1&,tk 1
and where P 2 k - = PZk= akfor 1 6 k < n. Now Proposition 2 immediately gives the result.
Plainly L (f o 17) = f for each f E C(Dm)since in this case F is constant modulo G . To show that, for each g, E A(E), y E A(Dm), we can confine our attention to the case where g,(x) = elnx if it can be proved that IlyllacDm)< C for some constant C not depending on n. But, in this case, F is defined by F(E) = einekfk and y(o) = yk(ok),where yk(0) ein ( t 2 k e l + t 2 k ) ) and yk(l) = (e1nf2k-1 einfZk ). For each func=+(I+ tion h : D + C, IlhllAcD,= Ih(0) h(1)l 4 Ih(0) - h(l)l. Hence IIY~IIA(D-) = I I Y ~ I I A ( D ) < 1 This last inequality follows Il~Ila(om)< from the observation that for any two real numbers a and b, lcos a cos bl + [sin a sin bl < 1; hence $11 + eia + elb + ei(a+b)l $ 11 - ela - eib + ei(a+b)l< 1, which implies IlykllAcD) < 1 for each k 2 1.
n?
n?
We first of all define a linear operator L : C(E) + C(Dm) such that, for each f E C(Dm), L (f 0 17)= f and for each f in A(E), L( f ) belongs to A(Dm). These two properties give the required implicatiorl (b) (a) and L will be used to show how problems of spectral synthesis can be transferred from E to Dm. Let P: Dm + Dm be the continuous homomorphism defined by dw, ~,,(mod2)for P(E) = coifs = ( E ~ )o~= ~( ~~ ~, ) ~ ~ , a n=e2,-, each k 2 1. The kernel of P is the subgroup G of Dm consisting fo all
+
n?
+
+
+ -
+
8. Spectral synthesis fails in A(Dm)
8.1. THEOREM VII : There exists a real function f in A(Dm) such that the closed ideals I, generated by f", n 2 1, are all distinct. The ideal I, is the closure in A(Dm) of the set of all products f"h, h E A(Dm); hence I,,, c I, for all n 2 1 and we have to show that f" # I,+,. It is sufficient to prcduce a pseudomeasure S,, i.e. a distribution whose Fourier coefficients form a bounded sequence, such that # 0 while for all h in A(Dm), (S,, hf"+ l) = 0.
(S,,f")
7.4. Proof of (b) => (a)
+
n?
+
(8.22)
.
For a better understanding of the proof, we present a candidate S, If
f:R -+ R is a regular function with a finite set F of zeros without multi-
x,,,
plicity, a distribution S1 = 6' of is well defined by (f'(1)l f'@)l)-' 6' (x - A) p, where 6' is the derivative of the unit mass concectrated in 0 and p is a measure carried by F. Hence (S, ,f = IAEr(lf'(A)l)-' > 0 while, for all regular h, (S, , hf 2, = 0. Unfortunately S, is not a pseudomeasure.
+
H i s defined in Section 7.2
>
ULTRA-THIN SYMMETRIC SETS
256
10. Counter-examples. A random set due to Salem 10.1. THEOREM IX: For each p > 2 there exists a sequence (tk),>1 of positive real numbers such that (tk+1/tk)' < i-co and such that the compact set E of all sums 1 E , = 0 or 1 has thefollowingproperties:
xk,
257
ULTRA-THIN SYMMETRIC SETS
Ekal
(a) E is a set of multiplicity; (b) there exists a complex valued bounded continuousfunction y : R -t C whose spectrum lies in E and which cannot be approached, in the topology of uniform convergence on compact subsets of R, by a bounded sequencey , k 2 1 , of trigonometric sums whose frequencies belong to E (here, bounded means: sup sup Iyk(t)l < + a).
negligible set N j . Let N be the union of all the N j ; the measure of N is 0 and for o # N a n d j 2 1, lim jf, < 1 which implies lim f,(o) = 0. k++m
k-+a,
To obtain the estimate (8.26), we apply the following lemma: 5: There is a constant C such that, for each integer m 2 1 and LEMMA each interval I of real numbers whose length III exceeds n/2,
,
k>l
teR
10.2. Let 0 < a < 3 be such that a p > 1. Each t, will be chosen at random and independently of the others in the interval I , = [(k!)-", 2 ( k !)-"I ; then (tk+ Itk)' < + co and for almost all choices of the sequence o of such t,, the corresponding compact symmetric set E ( o ) o all z y e k t k ,E , = 0 or 1 will have the required properties. Let Q be the compact product space I,, dt, the Haar measure of I , normalized by jIkdt, = 1 and d o the product Radon measure on Q: d o = dtl Q ... @ dt, @
xk2
nk,
a * . .
10.3. To prove that E ( o ) is a set of multiplicity, we use the function y ( x , o) = Icos xtkI as we did in Chapter 111, Section 3.3). If ol is sufficiently close to 5, which may always be assumed, E ( o ) is contained in [0,n[ and to prove that y ( x , o)vanishes at infinity, it suffices to consider only integral x. Let F,, k 2 1, be the finite set of integers x such that 2n(k!)" < x < 2n ( ( k I)!)" and letf k ( o ) = sup y ( x , a).To prove that lim f,(o)
n?
+
XPF~
k-
= 0 for almost all o in Q, the following lemma will be needed.
+m
Proof: By pericdicity afid symmetry the problem can be reduced- ~to2 / the 2. in case where I = [0,n/2]. For a suitable I > 0 , we have cos x < e [O, 11 and for any I, cos x < cos 1 in the remainderinterval [I, n/2].Hence cosmx dx < e-mx2'2dx + (42) cosmI Jn/2m,whichgives(8.27). The integral of a product of independent random functions is the product of the integrals. Hence j, ym( x , m) d o = jIklcos x tklmdt, < ,j lcos xtjlmdtj. If x E F,, each of the integrals in the last product is bounded by cm-lI2. Therefore ym ( x , co) d o C Clearly jDf :(a) d o C ym ( x , a) d o C (Card F,) c k m - ' I 2 < 2n ( ( k + I)!)" ~ , m - ~ Choose '~. m = m , = k and then for X > 1 we have Xmkc k m k ( ( k + I)!)" < + co which, by Lemma 4, finishes the proof. To prove the remaining statements of Theorem IX, the additive properties of the random sets E ( o ) must be described.
-
Jr2
nt
ny
l, xxsFk SO
xk,
10.4. Additive properties of the random symmetric sets E ( o )
DEFINITION 4: Let (tk)k,l be a sequence of real numbers such t h a t x y Itk( < + co. We say that (t,),, is fully independent if for any bounded sequence ( P , ) , ~ of rational integers the following implication holds:
LEMMA 4: Assume that an increasing sequence m , of integers tending to injinity exists such that for each X > 1
Then,for almost all o in Q, lim f,(co) = 0. k-r
+ co
x,,
Proof of Lemma 4: Let gj(w) = jmk f ,m" (a);g j ( o ) E L1 (Q, d o ) and so each gj is finite almost everywhere, i.e. for all o not belonging to a
It is easy to show that full independence implies lim -t,+,/t, = 0 but a Q-linearly independent sequence (tk)k21of real numbers such that lim tk+l / t k = 0 is not always fully independent. The terminology of the following proposition is that of Theorem IX: I , = [(k!)-", 2(k!)-&]and o = (tk)krl is a random choice of t , in I,. PROPOSITION 5: For almost all o in Q, (t,),,
is fully independent.
258
ULTRA-THIN SYMMETRIC SETS
ULTRA-THIN SYMMETRIC SETS
Proof: Let N be the set of all o in SZ for which there is a bounded sequence (p,),. of rational integers such that x y pktk = 0 ;if furthermore, such a sequence (pk),,, exists with lpkl < m for all k > 1, we write o E N(m) : Nis the union of all such N(m). If o E N(m),there is a smallest value j of k for whichp, # 0 and we then write o E N (m,j ) : N(m) is the union U N (m,j). If all these N (m, j ) have measure zero, so does N acd
,
1
j> 1
Proposition 5 is proved. Each N ( m , j) is a closed subset of 52 and the measure of N ( m ,j) will be computed by means of Fubini's theorem. Each o in N (m,j ) can be written as a pair ( o r ,a"),where of= ( t , , ..., tj) and o " = (tj+,, ..., t,, ...). If for each fixed o" in Q" = j+ I, the measure, computed in = Q f , of all of such that (of, o") E N (m,j)) is 0, so is the measure of N (m,j). But the statement ( a ' , a") E N (m,j ) is one of the 2m statements
nl,k,j~k
nk,
I
/
xk,
where G(o")is the set of all sums j+ pktk for all integral pk such that IpkI < m. Since tk+1/tktends to 0. the measure of the symmetric set G(otf)of real numbers is 0 and so is the measure of the set of tj satisfying (8.28).
El,@
< CIISlla*
and S,
-+
S in o (W*, 9). (8.30)
We can state PROPOSITION 6: For almost all o in Q, A (E(o))is of type 2. This at once finishes the proof of Theorem IX: in the situation of restriction algebras A(E) to compact sets E of real numbers, (8.30)is equivalent to $,(x) -+ $(x) uniformly on compact subsets of R (8.31) and
Then for almost all o in Q, the following equality holds for each complex measure p carried by a finite subset of E ( o )
xl,@
Before we conclude the proof, we shall describe the difference between the Banach algebras A(E) and A(E,). It is unknown whether a restriction algebra to a set of multiplicity can be isomorphic to a restriction algebra to a set of uniqueness. Such a theorem would prove that for almost all a E Q, A ( E ( o ) )and A(E,) are not isomorphic. The method we shall use is the following: let 9 be a regular semisimple commutative Banach algebra which will be identified, by means of Gelfand's representation, with a function algebra on the spectrum K of 9 ;the pseudo-measures are the elements of the dual space W* of W and the support of such a pseudomeasure can be defined in a natural way. Let o (W*, 92) be the weak-star topology. We say that 92 is of type 1 if each pseudo-measure S is the weak-star limit of a sequence S,,, n >, 1, of pseudomeasures carried by finite subsets of K. If this is not the case, we say that 9 is of type 2. General theorems of Banach give the following information on Banach algebras of type 1 :there is a constant C > 0 such that for each S in W*, a sequence S, of finitely supported pseudomeasures can be found such that
IISnll,*
,,
COROLLARY : For B > 3, let ( s ~ ) be ~ a fully independent sequence of real numbers chosen in ~ ( k ! ) -2(k!)-'1 ~, and let El be the ultra-thin symmetric set of all sums xk,l E ~ S L E, , = 0 or 1. Let H be the homeomorphism from E ( o ) to El defined by
Proof: Let 8 be the additive group generated by E : @ is the set of all sumsxk>,pktk corresponding to all bounded sequences (p,),. of rational integers; let 8, be the additive group generated by E l . The full independence of the sequences (t,),, and ( s ~ implies ) ~ ~that Hcan be extended by H (Xkalpktk)= Yk.,pksk to an isomorphism, still denoted by H, from 8 to (3,. If P is any finite trigonometric sum, P(t) = c(1) exp 2nih and if G(t) = c(A) exp 2niH(iZ)t, Kronecker's theorem (Chapter I, Section 5) gives 11 Pll oL, = 11 Q 11 m, which is (8.29).
10.5. Last part of the proof of Theorem IX
as we have already observed in Chapter VII, Section 3.2. The following lemma will be useful in proving Proposition 6: i
1
LEMMA 6: Let f , : R -+ [O, 11 be defined by f,(x)
=
cos t l x ..-cos tk-,x sin tkx cos t,+,x
Then for almost all o in 52, sup I fkl -, 0. R
cos t,x
..-
260
ULTRA-THIN SYMMETRIC SETS
ULTRA-THIN SYMMETRIC SETS
The proof is an easy modification of that given in Section 10.3: the term sin t k xinstead of cos t,x ensures uniform convergence to 0 on compact subsets of R. Note that f , = for a suitable measure p, carried by E.
THEOREM X: For almost all o in 9 , A ( o ) is not a coherent set of frequencies. + t,) x c o s n t , ~ c o s n t , ~ ; Proof: Let P, ( x , o) = exp n i ( t , + the spectrum of P, lies in A and Theorem X will be proved if, whenever 0 < a < b there exists a negligible subset N (a, b) of SZ such that, P, ( x , o) converges uniformly to 0 on [a, b] for o 4 N (a, b); then, if o does not belong to the negligible union of all the N ( j - l , j), j 2 1, A ( o ) is not a coherent set of frequencies. Let F, be the finlte set of all x = j (kt,)-l, j E N , lying in [a, b]. For each x in [a,b ] ,there is a y in F, such that Ix - yl < (2ktk)-I and Bernstein's inequality (Chapter V, Section 4) shows that IP,(x) - P,(y)( < C k - l , where C does not depend on k . Hence it suffices to prove that f k ( o ) = sup IPk (x, o)I tends to 0 for almost all o in 9.The remainder of
LEMMA 7: The pseudo measure norm of H(p,) does not tend to 0. Proof: By Theorem 11 this norm is greater than C-l sup cos x , . . cos x k - sin x , cos xk+ ...,where C is a positive constant and ( x , ) ~ , runs over all sequences of real numbers. But this last sup is 1. Proof of Proposition 6 : Assume that A (E(o)),which we shall write A(E), is of type 1 and that o does not belong to the union of the two negligible sets which arise in Lemma 6 and Proposition 5 respectively. Let (,u,,~)~,,be a sequence of measures carried by finite subsets of E and satisfying IIpk.jllpM < CllpkllpMand (pk,j , f ) -, ( p k , f )for all locally constant functions f on E; such a sequence exists by (8.30). Then by Corollary of Proposition5 ( [ H ( , ~ ~ , = j ) ( (l p~k~, j ( ( p M acd clearly (H(P,,~), f ) -, (H(pk),f ) for all locally constant functions f on E l . Hence H(p,) is the weak-star limit of the bounded sequence of pseudomeasures H ( p k S jj) , 2 1, since the locally constant functions are dense in A(El), and since norms decrease in the passage to weak-star limits l l H ( p k ) l l< ~ ~lim IIH(pk,j)ll < Cllpkllp~.Lemmas 6 and 7 show that
,
j++m
this is not the case.
11. Other random sets Let (tk)k,l be an increasing sequence of positive real numbers such that t,,, 2 3tk (for all k 2 1 ) and let A be the set ofall finite s ~ m s z ektk, ~ , ~ E , = 0 or 1. Theorem IV asserts that if (tk/tk+l)2< + co,A is a coherent set of frequencies. We shall prove that the exponent 2 cannot be replaced by a greater one and that a sequence (t,),,, can be chosenin such a way that (tk/tk+ < while A is not harmonious. In both cases random methods will be used. Let 0 < ol < & let I,, k 2 1 , be the interval (k!)" < x < 2(k!)", let dt, be the Lebesgue measure of I, normalized by j,, dt, = 1, let SZ be the compact product space I, and let d o be the product Radon measure on SZ defincd by d o = 8 dt,.
I,,,
xkg
+
nk,
k b1
For each o = (t,),, in SZ, let A(@)be the corresponding discrete set of sktk,E , = 0 or 1. all finite sums
xk,
26 1
xeFk
the proof is an easy transcription of that given in Section 10.3. If 4 < ol < 1 the symmetric set A ( o ) constructed with (tk)kal,where tk E I, = [(k!)",2(k!)"]is a coherent set of frequencie?. Moreover for each positive E , there exists a T (depending on E and co) such that for each trigonometric sum P whose frequencies belong to A , sup /PI < ( 1 E )
+
R
sup IP(x)(.This property is shared by all harmonious sets. However 1x1
=
THEOREM XI : For 3 < ol < 1 , let I, be the interval [(k!)",2(k!)"]and let o = (t,),,, where each t , is chosen at random and independently of the other tj in the interval I,. Let A ( o ) be the discrete set of allJinite sums cktk,E , = 0 or 1. Then for almost all choices of the sequence of t k , k 2 1 , A ( o ) is not harmonious. (To be more precise we recall that each t, is equidistributed in I,.)
xk>l
This theorem is very sharp since, following the method given in Chapter 11, Section 4, it can be shown that i f T k , , tk/tk+ < co A is harmonious. We shall prove that, if 0 < a < b there is a negligible subset N1(a,b) If o does not of SZ such that, for o 4 N (a, b), inf sup lllxll 2
+
A.
a<x