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0,
II Dl (f
* Xn) II X
< M nY
II f II X
(f E X; n E JI?) •
can be appfLox-i..mated by .the convolu.tion -i..ntegfLai
that
(2.21)
60fL ~ome
PROOF:
II f
0
-oo
Hence for arbitrary
h E (-1,1) and integral
m > 2
(2.24)
Since
Uk belongs to
w~ for every
k > 2, one has by (2.11)
1 < 6 (1 - h) II 0 Uk II X •
(2.25)
Since the convolution product is commutative and associative, one can rewr i te
Uk as
(k = 3,4, ••• ) •
Uk = (f-f * X k-l) *X k - (f-f*X )·X k-l 2 2 2k 2 This implies by (2.25),
(2.20) and (2.21) that
~ M(l - h)2 ky (1-a.),
which is also valid for
(2.26)
k =2. This yields by
(2.24) and (2.22) that
L
(-1 0 n n n
(3.1)
THEOREM 1:
pEN p+ theJte hold.6
FOIL aLe.
a)
x E [-1,1], Pn~(O)=1} (n EIP).
for
n
n
D, and
xm+1
(3.2)
whelLe
m = [(n + 1)/2 b)
fOIL
each even
n
E
P
i.6 the lalLge.6t 1L00t 06
thelLe exi.6t.6 a unique Pn
E
P + . m 1
N P~
.6uch
that P~ (1)
(3.3)
c) FOIL
n
E
IN odd :thelle ex.i.6:t.6 no
d) Fait each
(3.4)
Pn
E
j (j + 1)
NP~
.........l..L.......;.......:.;:"""'2
2 (2n + 1)
and each
< 1 -
P~
j
P E
n
E
IN thelle hold.6
(j) < 72j (j + 1) (1 -
the Itight hand inequality being valid nOlL all inequality only 601!.
PROOF:
N P+ n .6Uch that (3.3) holM.
n E
P~ (1)
,
P, :the le6t
hand
n ~ No = No(j).
First we need the Gauss-Jacobi mechanical quadrature formula
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
(see e.g. [19, p. 741, [57, p. q2k-l E P2k-l'
89
47;(15.3.2}]l. It states that foral!
k E :IN, -there holds
(3.5)
where with
Xj ,k'
1.2. j .2. k, denote the roots of the Legendre polynomial P
-l<xj+l,k<Xj,k !P!(I; )\(1- cos (1Tj(2n +1») > Ip!(I;)/
']
P! (1)
P! (I; ) ]
n
n
J
]
j (j + 1) /2,
the proof
n
(2n + 1)
of
d)
2
is com-
0
The polynomial
Pn of (3.10) satisfies for even
n
E
:IN
the
same extremal property as does the trigonometric Fejer-Korovkin kernel. Therefore one may justly call the kernel (Legendre-) Fejer-Korovkin kernel. If cisely the polynomials
n
Pn of (3.10); if
K
n
of
is even, the n
(1. 22)
Kn
are
the pre-
is odd then Kn(x) =Pn-l(x).
Concerning the approximation behaviour of the associated Fejer -Korovkin convolution integral we have
92
BUTZER,STENS and WEHRENS
by (1. 22) . One ha.t.
*
-
f E X
L fllx .::. 24 wl (xN;f;X)
a)
IIf
b)
f E LiPi(Cli X) -lif
c)
The
X with o~de~
K n
6OIl.
*
(n E P) .
2Cl Kn - f II X = O(n- )
~onvolution integ~a.l
06
Feji~-Ko~ov~in
(n ... 00; O0),
respectively, the following
COROLLARY 4:
one.
ha~
a)
60lt any
Folt ;the.
~ingu,ealt
int:e.gJta£. f
* v[ Bnl , n'
f E X, 0 < B < 1,
r E lP
o(n- r - Cl ) f
*
Kn ,
f E
(n'" "'; 0 < Cl
x, one.
~ 2) •
h4~
(n'" "'; 0 < a. < 1)
(n ... 00; 0
X,
then (L, TW) is Hausdorff. ~
p [w(x)f(x)] < d,
0, form a basis of a-neighborhoods
for (L,T ). W (iii) If W is non-vanishing at X, then, for each x E X, ox: L defined by
->
E,
0x(f) = f(x), is continuous and linear.
2. EXAMPLES (i)
If
L = C(X,E) and W is the set of F-characteristic func-
tions of compact subsets of open topology in (ii) If
X then
TW
is the
compact-
C(X,E).
L = Co(X,E), the vector space of all continuous func-
tions from then
TW
X to
E which vanish at infinity, and W = {l},
is the uniform topology in
In particular, if
Co(E,X).
X is compact, we get Co(X,E) =C(X,E).
(iii) If
X is locally compact, L = C (X,E), the vector space b of all bounded continuous functions from X to E, and
W = Co(X,F), then
TW
is called the strict topology
in
Cb(X,E) . We notice that in all these cases
W is non-vanishing
at
X,
and that, in the first two cases, W is directed.
3. THE BOUNDED CASE OF THE NON-ARCHIMEDEAN BERNSTEIN-NACHBIN PROBLEM Given a subalgebra A of
C(X;F) and a Nachbin space
the non-archimedean Bernstein-Nachbin problem consists in describing the closure of a vector subspace that is, such that For ~(x)
=
which is a module over
A,
AM C M.
x, y E X,
~(y),
MeL
for every
we say that ~
x:: y
E A. We denote by
(mod A) X/A
if and
only
the quotient of
if X
NON-AACHIMEDEAN WEIGHTED APPROXIMATION
by this equivalence relation. Each
wi y
of X, one easily sees that
Y EX/ A
is a set of
123
being a closed
L! y - weights, and then
(L I y , TW Iy) is a non-archimedean Nachbin space. M
loeal-tzable undelt A in (L,T W ) if the
is then said to be
TW - closure of
M in
cides with (or, equivalently, contains) the set
£A(M)
that
in
fly The
belongs to the lte~~It-te~ed
T
closure of
Wly
Mly
Lly}.
non-archimedean Bernstein-Nachbin problem
ea~e
con-
holds.
of the non-archimedean Bernstein-Nachbin prob-
lem occurs when the functions in every function in
L coin-
{f E L such
sists in asking for conditions under which localizability The bounded
subset
A are bounded on the
support
of
W.
We have then the following:
THEOREM 1:
In
bounded
~he
loeal-tzable undelt A -tn
PROOF:
ea~e,
evelty
A-module
eon~a-tned
-tn
L
-t~
(L,T ). W
The proof is an adaptation of the one given by Nachbin ([4])
in the real case, and is based on the following
LEMMA:
Le~
A be a
w-t~h ~he ~upltemum
~e~
and
00
un-t~alty e.e.o~ed ~ubalgeblta
noltm. 16, 60lt evelty
X wh-teh -t~ d.t~jo-tn~ 6ltom
~l""'~n E A
(i)
~iIKZ.
06
Cb(X,F)
Y EX/A, Ky
.t~
y, ~hen ~helte ex-t~~
a
equ-tpped eompae~~ub
Zl'."'Zn EX/A
~ueh ~ha~:
i = 1, ... ,n
= 0
~
(ii)
II~ ill
OQ
sup I~· (x) ~ XEX
I
n (iii)
~
~i (x)
= 1,
x E
~ 1,
i
*
1, ••• ,n
x.
i=l PROOF OF THE LEMMA: X
and
S: Cb(X,F)
Let ~
BX
be the Banaschewski compactificationof
C(SX,F) the isometry such that
S(f) = Sf is the
124
CARNEIRO
only continuous extension of of A under
f
SX (see [5 I ). Then
to
S, is a unitary closed subalgebra of
the quotient space
SX/SA
and
n : eX
G
+
SA, the image
C (SX,F). If
G is
the canonical projection,
then G is compact, Hausdorff, and zero-dimensional (see I6 1). Now, for
y n X ,
Y E G, we have that
this case, Y n X E X/A. Then
¢ if and only if
Ky n X
is a compact subset of X, which y E ~ (X)
Y n X. This implies that
is disjoint from
the finite intersection property, there exist
n r:2
that for
n (K y n X) = ¢. Putting i
i-I
= 1, ...
i
(i)
hiln(K
(11)
IIh i II n 2:
00
zi
hl, ••• ,h
o
)
n X
= y.
~
~
1f
(K y n X) = 0, wl, ..• ,WmE W
Y E X/A, there exists
j =l, ... ,m,
gy E M
Iwj(x) IlIf(x) - gy(x) II < e.
m U {x E X; Iw. (x) I IIf(x) - gy(x)1i > e} is comj=l ) pact and disjoint from y. By the Lemma, we can find Zl""'Zn EX/A
The set
Ky
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
and gi
0, pEr, wI' ... 'W
£
by hypothesis, gEM
which proves that
E W, x E X,
p ( f (x) - 9 (x)]
such that
k = 1 + mjx Iwj(x)l. Then
m
X/A are single-
p[wj(x)(f(x) - g(x»]
O.
f(x)
f
E
= f(y). a!!.e
al.6o
belongs to
such that
9 (x)
= 0,
A,
the
9
E
A
'I O.
0,
Since
implies
(i)
9
Y.
is
also
hl y = fly. (If 1.=0, h=O
I
w, Iwj(x}
E
'I
in
Ih(x) -f(x) I ==0 locaU.y compact,
(c) 16 f E
cb (X,F),
then
f (x)
fly) •
f E Co (X,F) , the.n
=
fIx)
b e.-
g E A, .[mpi.[e..6
0
f (x)
A i...6 a .6ubalge.bJta 06
f (y) .
and
C (X,F) b
be.iongl.> to the. I.>tJt.[ct ciol.>uJte 06
f
f
.[6
A .[6 and oniy
g E A, .[mpUel.>
g(y) , 6 OJ!. e.ve.lty
=a
fIx)
g E A, .[mpi.[e.-6
60Jt e.ve.Jty
A .[.6 a -6ubaige.bJta 06
16
g E A, .[mpUe.-6
0, 601t e.ve.Jr.tj
.[6 and
A
a nl!f .[ 6 g(x)
(i)
(ii) g(x)
0,
60Jt e.ve.Jttj
g(y),
g E A, .[mpli..e.1.>
g E A, .[mpl.[e.1.>
60Jt e.ue.Jty
=
f(x)
O.
= f(y)
f(x)
•
5. DENSITY IN TENSOR PRODUCTS If E, then
Sand Tare, respectively, vector subspaces of C (X, F) and S 0 T
the form
x
->
denotes the set of all finite sums of functions sex) t, with
s
E
S, t
E
T. Similarly, if
are zero-dimensional Hausdorff spaces, and tive1y, vector subspaces of denotes the set of all finite
THEOREM 4:
C(X ,F) and 1 sums
of
since
A 0 E
C(X ,F), 2 the
A 0 E
is an A-module, and (A 0 E) (x)
A is non-vanishing at
Corollary.
and
S2
functions
X.
i...6
= E,
and
are,
X 2
respec-
then
16 A i..1.> l.>epaJtat.[ng and non-uani..l.>hi..ng on X,
and i..6 we. Me. i..n the. bounde.d cal.> e., the.n
PROOF:
Sl
Xl
of
of
the form
A 0 EeL,
Tw-de.Me. i..n
for every
L.
x E X,
It suffices then to apply Theorem 1,
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
129
COROLLARY 1: (i)
C ex, F)
® E
i.6 del'!.6 e in
C (X, F),
60ft the compact
-
open
.topology. (E)
16
X i.6 locally compac.t, K (X,F) ® E
60ft .the u.ni60ftm .topology.
(K(X,F)
i.6 den.6e in Co(X,E),
i.6 .the .6e.t 06 aU
con-
tinu.ou..6 .6calaft 6u.nc.tion.6 wi.th compac.t .6UPPOft.t) . (iii) 16
X i.6 locally compac.t, Cb(X,F) ® E )..6 den.6e in
S,(X,E),
60ft .the .6.tftict topology.
COROLLARY 2 (Dieudonne): (i)
(C(Xl,F) ® C(X 2 ,F»
® E
i.6 den.6e
~n
C(X
x X ,E),
l
2
.the compact-open topology. (ii) C(Xl,F) ® C(X 2 ,F)
i.6 den.6e in
C(X
X2 ) ® F.
x
l
6. EXTENSION THEOREMS
THEOREM 5:
r6
E i.6 a non-aftchimedean Fne.che.t .6pace oven
F, and
Y
i.6 a non-emp.ty compac.t .6u.b.6e.t 06 .the zeno-dimen.6ional Hau..6doft66 .6pace X,
then evefty
E -valued continu.ou..6
a bou.nded continu.ou..6 6u.nc.tion on
PROOF:
6unc.tion on
can be extende.d .to
Y
X.
We wi 11 employ a technique due to De La Fuente [ 7 I
linear mapping
Ty: C(X,E) ~ C(Y,E), defined by
Ty(f)
=
fly
S C C(X,E), denote
uni tary subalgebra of
Ty(S)
by
C (y ,F), and
Sly. Then
A = Cb(X,F) Iy
M = C (X, E) I y b
Since the constant functions belong
to
By Theorem 1, Corollary,
is dense in
Assume first that
Cb(X,E) Iy
M, M(x)
X is compact. Then C (X, E)
space, and so is its quotient by the closed subspace
is an
is spaces.
clearly continuous for the compact-open topologies in both For
The
is
a
A - module.
E, for each x E y. C(Y,E). is
a
Frechet Now
130
CARNEIRO
we claim that C(X,E) Iy,
C(X,E)/K
is linearly and topologically isomorphic to
for which it is enough to prove that
homomorphism. Indeed, given
U,
is a topological
a basic neighborhood of 0 in
then
U
{g E C(X,E); p[g(x)]
O.
neighborhood
V n [C(X,E) Iy ],
it is enough to prove the reverse inclusion. Let then with
g E C(X,E). Then
joint from
Y.
that
0 on
is
.p
G = {t EX;
By ultra-normality of
is such that
G, 1
fEU
on Y, and
and
Ty(f)
Therefore, C(X,E) Iy = Cb(X,E) Iy
X,
given
g = hlx
THEOREM 6:
16
is complete, and
x, then
tion in
PROOF:
thus
h E Ty(U) .
closed
in
Cb(X,E) Iy = C(Y,E). SFX
the Banaschewski compact-
h E C(SFX,E)
such that
Then,
f =hl y • '!he
is the required extension.
E -il.l a non-altc.himedean Fltec.het I.lpac.e ovelt
il.l a c.!ol.led I.lubl.let I.lpac.e
on X. Then f=.pg E C(X,E)
By the previous result, C(Y,E) = C(SFX,E) I y .
f E C(Y,E), there exists
function
.p EC(X,F) sum
h, which proves that
Now, in the general case, take X.
is compact and dis-
there exists
l.p I 2. 1
C(Y,E). Since it is also dense, we get
ification of
> d
p[g(t)]
06
F, and
Y
the zelto-dimenl.liona! !oc.a!!y c.ompac.t Haul.ldolt66
evelty 6unc.tion in
Co (Y ,E)
c.an be extended to a 6unc.-
Co(X,E).
We omit the proof, which is similar to that of Theorem 5.
REFERENCES
[1]
A. F. MONNA, Ana!Yl.le non-altc.himedienne, Ergebnisse cier Mathematik und ihre Grenzgebiete, Band 56, Springer-Verlag, Berlin, 1970.
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
[21
L. NARICI, E. BECKENSTEIN and G. BACHMAN,
and
Vatuat~on
Theo~y,
131
Fun~t~onat
Anaty~~~
Pure and Applied Mathematics,vol.
5, Marcel Dekker, Inc., New York, 1971. [31
J. P. Q. CARNEIRO, Ap~ox~ma~ao Ponde~ada nao-a~qu~med~ana,{Doc toral Dissertation), Universidade Federal do Rio de Janeiro, 1976; An. Acad. Bras. Ci. 50 (1978), 1 - 34.
[41
L. NACHBIN,
We~ghted App~ox~mat~on
Cont~nuou~
Fun~t~OM:
6M
Reat and
Atgeblta~
and
Set6-Adjo~nt
Modute~
06
CMe~,
Comptex
Annals of Math. 81 (1965), 289 - 302. [51
G. BACHMAN, E. BECKENSTEIN, L. NARICI and S. WARNER,
Rings of
continuous functions with values in a topological field, Trans. Amer. Math. Soc. 204(1975), 91-112. [6
1
J. B. PROLLA, Nonarchimedean function spaces. To appear L~nealt
Spa~e~
App~ox~mat~on
and
in:
(Proc. Conf. ,ObeTh'Olfach,
1977: Eds. P. L. Butzer and B. SZ. - Nagy), ISNM
vol.
40, Birkhauser Verlag, Basel-Stuttgart, 1978.
[71
A. DE LA FUENTE, Atguno~ Ite~uttado~ ~oblte apltox~ma~~on de 6un~~one~
ve~tolt~ate~
t~po
teoltema
We~elt~t~a~~-Stone,
Doc-
toral Dissertation, Madrid, 1973. Etement~
L. NACHBIN,
[91
J. B. PROLLA,
06
Appltox~mat~on
Theolty, D. Van Nostrand Co. Inc., 1967. Reprinted by R. Krieger Co. Inc., 1976.
[81
Appltox~mat~on
06
Ve~tolt
Vatued
Fun~t~oM,
Holland Publishing Co., Amsterdam, 1977.
North-
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. ProUa (ed.) ©North-Holland Publishing Company, 1979
TH~ORIE
SPECTRALE EN UNE INFINIT~ DE VARIABLES
JEAN-PIERRE FERRIER Institut de Mathematiques Pures Universite de Nancy 1 54037 Nancy Cedex, France
1. L'utilite d'une theorie spectrale et d'un calcul fonctionnel holomorphe en une infinite de variables a ete mise en lumiere par la recherche de conditions d'unicite pour Ie calcul fonctionnel holomorphe d'un nombre fini de variables et des algebres (cf (21).
Disons, de faQOJ1 schematique,
a
spectres non compacts
que l'unicite est
etablie
pour undomaine spectral pseudoconvexe et en particulier polynomialement convexe et que, d'autre part, tout domaine de
~n peut s'inter-
preter comme laprojection d'un domaine polynomialement convexe, mais d'un nombre infini de variables. fa~on
De
a element
classique, etant donnee une algebre
unite (toutes les algebres seront supposees desormais telles),
on se donne des elements ora) de de
([:n
A, commutative et
a
=
a , ... ,an l
de
A et on definit
(al, ... ,a n ) comme l'ensemble des points
tels que l' ideal engendre par
a
l
s
Ie spectre
=
- sl' ... , an - sn
(sl' ... ,sn) soi t pro -
pre, plus precisement comme Ie filtre des complementairesdes parties S, dites spectrales,
sur lesquelles on peut trouver
des
fonctions
2. Pour decrire une situation semblable en dimension infinie, il est nature 1 de remplacer
en
par un espace localement convexe
donnee de
par celIe d'une application lineaire bornee
133
E
et la
a
134
FERRIER
du dual
E'
de
E dans
A.
a
La notion de spectre correspond alors systeme fini
0, I{)
algebre."l.~
sur une
~, dent
de fonctions sur
l'injectivite n'est malheureusement pas claire. 5'il n'y
a
pas
de
probleme dans Ie cas d'un produit, la situation n'est pas dans Ie cas d'un produit fibre sur un domaine de nier est pseudoconvexe (cf [1 1,
[2
~n, sauf si ceder-
1)•
4. De1aissant ici 1e probleme de savoir si 1es fonctions holomorphes sur
du cal cuI fonctionnel sont des fonctions, concentrons-nous
spectre et cherchons si on peut remplacer dans certains cas Ie
nI{)
teme projectif des
par un domaine
a
pouvoir connaitre des familIes (SI{)
n
des parties
de
SI{)
de
0, est specn
n
trale pour a ? If faudrai t pour cela que pour un element de E', c'est ait
I{!(n)
a dire
E cr(al{!)'
une suite (X ) de
(4)
~ X
n
a n
de la sphere unite
I Xn I
= 1
on
dire
S
n
cr (~X a ) , n n n
E
et avec uniformite par rapport a (X ). n En effet, s'il existe E > 0 tel que contient la boule ouverte boule ouverte
~
II «[:) telle que
n
c'est
I{!
I{!(~n
B(Zn,E) et
E , alors
=
°
B (~ Xn Zn' E) de sorte que
I{! (n) ••
contient (~X n Z n ) -> E.
5. On peut done se poser de fa90n generale Ie probleme suivant:etant donnee une suite bornee (an) de que
AN et une suite
(Sn) de
telle
G:!N
Sn E cr(an) avec uniformite par rapport a n , est-ce que l'on
la relation (4) pour toute suite (X ) de n avec uniformite par rapport
a
II (G:!) telle que
~IX
n
a
I = 1,
(X n ) ?
Considerons Ie cas particulier d'une algebre
de
Banach.
On
verifie tout d'abord, en prenant des caracteres, l'inclusionsuivante, dans laquelle
sp(a n )
Sn est remplace par l'ensemble
tersection du filtre
(qui est l'in-
cr(an»
~
n
X sp(an):J n
sp(~
n
Xna n ).
Cette meme inclusion montre done que pour tout choix de Sn E cr(ad' on a la relation (4). Cependant il resterait
a etablir
l'uniformite
TH~ORIE SPECTRALE EN UNE INFINITIO DE VARIABLES
137
par rapport au choix d'une suite (An) de la sphere unite de II n' y a pas de difficul te si on remplace la borne sur les coefficients avec
e: >
u
i
a
la distance
par Ie fait que
5 contienne un E-voisinage du
fixe. En effet si
a
A
AE
designe 1 'ensemble des points dent
est strictement inferieure
2:A
n
(sp(a»E
n
On est ainsi conduit
a
spectre
a
E on a
(2: A sp (a » (; .
n
n
etudier la croissance des
coefficients
spectraux en fonction de la distance au spectre. Dans un sens on a l'inegalite:
qui s'etablit facilement en prenant que
IX (u i ) I 2.
t
x(a) E sp(a) et en
sachant
II ui II •
La question fondamentale concerne l' autre sens: peut - on tout
E > 0
trouver une borne des coefficients u i (s) avec qui soi t independante de a, II a II < 1 ?
pour
d(s,sp(a»~E
BIBLIOGRAPHIE;
[1
1
J .-P. FERRIER, Theorie spectrale et approximation par des fone-
tions d'une infinite de variables, Coll. An. Harm. Complexe, La Garde - Freinet 1977. [2 1
K. NI5HIZAWA, A propos de l' unic! te du calcul fonctionnel holomorphe des b-algebres, these, Universite de Nancy, 1977.
[3
1
L. WAELBROEK, Etude spectrale des algebres completes, Acad. Roy. Belg. Cl. 5ci. Mem., 1960.
This Page Intentionally Left Blank
Approximation Theory and FUnctional Analysis J.B. Prolla (ed.) @North-Holland Publishil1{J Company, 1979
MEROl-10RPHIC UNIFORM APPROXIMATION ON CLOSED SUD SETS
OF OPEN RIEMANN SURFACES
P. M. GAUTHIER* Departement de Mathematiques et de Statistique Universite de Montreal, Canada Dedicated in memory
of Alice Roth
1. INTRODUCTION Let
F be a
face R. Denote by
(relatively) closed subset of an open Riemann surH(F) and
M(F) respectively
the
holomorpl1ic
and
meromorphic functions on (a neighbourhood of) F. Let A(F) denote the functions continuous on
F and holomorphic on the interior
F
O
of F.
Recently, the problem of approximating functions in A(F) uniformlyby functions in H (R) has been considered by Scheinberg [17 I . In the present paper, we consider the problem of approximating a given function on
F uniformly by functions in H(R) and obtain, as
a
corollary,
a
result related to Scheinberg's. Our method of approximation is based on the technique of the late Alice Roth [15J. We shall rely on Scheinberg [17 I for some results
on the
to-
pology of surfaces. Without loss of generality, we shall assume that every Riemann surface its closure in of
R if
R is connected. A subset is bounded in
R is compact. A Riemann surface
R'
is an
R
if
ex~en~ion
R is (conformally equivalent to) an open subset of
R'. If
* Research supported by N. R. C. of Canada and Ministere de l' !;ducation du Quebec. 139
140
GAUTHIER
furthermore
R
'I R', R' is an e.6.6ent..i.al
that a closed subset a
exten.6ion of R. We shall say
R is e.6.6 entiatty 06 6inLte 9 enu.6 if F has
F of
covering by a family of :pairwise disjoint open sets, each
nite genus. Denote by morphic on its on
the uniform limits on F of functions r:rero-
M(F)
R with poles outside of
F and by
F of functions holomorphic on
compactification of
of fi-
if (F)
the uniform lim-
R. R* will denote the one point
R.
The central problem in the qualitative theory of approximation is that of approximating a given function on a given set. In thisdirection we state our principal theorem.
(Loc.atiza.tion):
THEOREM 1:
Let F be c.to.6ed and eMentiaUy 06
nite genu.6 in an open Riemann .6ufL6ac.e M(F)
R.
Then, a 6unction
f
i.6
6iin
i6 and onty i6
f I K n F
(1)
60fL evefLy
c.ompac.t .6et K in
E M(K
n F) ,
R.
If we drop the condition that
F be essentially of fini te genus,
then the theorem is no longer true [9 ). dition, for
However, we may drop the con-
R planar, since it is trivially verified by all
F.
In
this situation, Theorem 1 is due to Alice Roth [15). An immediate consequence of Theorem I is the following
Walsh-
type theorem, which was first obtained for planar R by Nersesian [141.
THEOREM 2:
Let F be c.to.6e.d and eMentiaUy 06 6inite genu.6
open Riemann .6ufL6ac.e
R.
A
.6u66ic.ient c.ol1dition 60fL
that
A(F n
V)
in
an
A(F) = M(F)
i.6
141
MEROMORPHIC APPROXIMATION ON CLOSED SlJBSETS OF RIEMANN SlJRFACES
60ft
eveJr.1j bounded open -6et
V -iI'!
R.
By the Bishop-Kodama Localization Theorem [12], the open sets
we may replace
V by parametric discs.
The following is a Runge-type theorem.
THEOREM 3:
Let
F
be c.[o-6ed and e-6-6entiaU.y 06 6inite genu-6
open Riemann -6uJr.6ace
R. Then
H (F)
C
M(F). MOJr.eov eJr. , H (F)
C
in
an
R (F) i6
and onllji6 R*\ F.[.o connected and .f.oc.a.f..f.y connected.
Recently, we proved Theorem 3 for more restricted pairs (F, R) [ 7] •
From Theorem 2, we have a corollary on Walsh-type approximation by holomorphic functions.
THEOREM A:
(ScheinbeJr.g [17]):
Let F
6inite genu-6 in a open Riemann -6uJr.6ace
A(F)
R(F) i-6 that
R* \ F
be c.f..o-6ed and
R.
e-6-6entia.f...f.1j
oS
A -6u66icient conditioI'! naiL
be connec.ted and .f.oca.f..f.1j connected.
Scheinberg actually obtained this result for somewhat nnre general pairs (F,R). For arbitrary pairs (F,R), the condition that R*\F be connected and locally connected is also necessary but
no
longer
sufficient [9]. In fact, Scheinberg has shown that there is no topological characterization of pairs (F,R) for which A(F)
PROOF OF THEOREM A:
Since
= R(F)
[17].
R*\ F
is connected, it follows from the
Bishop-Mergelyan Theorem [2 ] that
F satisfies the hypotheses of The-
orem 2, when the sets f
> 0,
there is a
V are parametric discs. Thus, if
gl E M(R) with
if(z) - gl(z)1
< E/2,
Now by Theorem 3, there is a g E H(R)
z E F.
such that
f E A(F) and
GAUTHIER
142
This completes the proof of the corollary. A closed set F in
R is called a set of Carleman
tion by meromorphic functions, if for each ti ve and continuous on
there is a g E
F,
I fez)
-
< £(z),
g(z)1
f E A(F) and each M (R)
THEOREr14:
£
posi-
with
Z E F •
The next result characterizes such sets completely when result is known for
a~7~)roxima-
F
O
{tf.
This
R planar [14] .
Let F be c.io.6ed w-Lth empty -Lntelt..i.o/t .i.n an open IUemaf'lf'l
6ac.e R. Then F -L.6 a .6et 06 CaILieman appILox.i.mat.i.on
by
6u.Jt-
meILomOlLph1c.
6unc.t.i.on.6 .i.6 and only 16
C(F n K)
601i. eac.h c.ompac.t
.0
et
M(F n K),
K.
2. FUSION LEMr1A Using Behnke-Stein techniques, Gunning and Narasimhan [11] have shown that every open Riemann surface R can be visualized in a very concrete way. Indeed,they showed that fication) above the finite plane
~.
R can be spread (without ramiTo be precise, they proved that
R admits a locally injective holomorphic function
p. Thus
is the spread. We wish to reconstruct the Cauchy kernel of Behnke-Stein on R, something resembling (q - p)
-1
. Conceptually
we prefer to think of p
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
and
q as both lying on
R, however, for proofs, it may be prefera -
ble to think of two copies
z and
the
Rp
p :
R
p
x R
We construct an
r ( .,
_
z) - 1
Rq of
Set on
~
x
z
R spread respectively above
~
;:;
..-+- (z , ;:;) •
cover of
o~en
Dq be discs about
~.
~
q
(p , q)
p
and
;:; planes:
p x
°overand
143
p and
q
R x R. If
(p,q) E R x R,
respectively which lie
U(p,q) = Dp x Dq • Consider the Cousin data U ( p,q. ) S ~nce .
R x R
schlicht
which
is
is Stein, the first Cousin prob-
lem can be solved. Hence there is a meromor"l?hic function whose singularities are on
let
on R x R
the diagonal. In the neighbourhood of
a
diagonal point, we have, in local coordinates (forever more given by p
x
p), that
1
t(l;;,z) -
I; -
is holomorphic. (1; , z) means
z
(p,q), where
pip) = I; and p(q) = z.
We shall persist in this abusive notation, since it is invariant under local change of charts within the atlas given by the function
a Cauchy kernel on
p x p. We call
R since
We shall now extend to surfaces the powerful Fusion
Lemma
of
Alice Roth [15] .
FUSION LEMMA: mann .6uJz.6ac.e
Let
K , K , and 2 l
R, w.i.th
Kl
a.nd
K2
K be c.ompac.t .6ub.6e.t.6 06 an open R-i.ed.i..6jo.i.nt. TheILe '£.6 a. p06.i.tive numbe!t
GAUTHIER
144
a .6uch tha.t.£6 .6a.t.£.6 nljil1g,
m l
a.l1d
m 2
Me. a.111j two me.ftomOftph.£c 6ul1ct.£011.6
011
R
E > 0,
60ft .6ome.
Im l
(1)
- m LK < 2
m, me.ftomoftphic
the.11 the.fte. i.6 a. 6uI1ctiol1
1m -
(2)
E,
R .6uch tha.t 60ft j = 1,2,
011
mj I K UK. < aE J
PROOF:
We may assume
bourhoods and
and U2 of Kl and K2 respectively such that l is precompact. Moreover, we may assume that the
U
R\ U 2
aries of
K2 \ K 'I ¢. Thus, we can construct open neigh-
U
l
curves. Let
and E
U
be the compliment of
U
l
U
U
in
2
(R \ U ) U K2 U K.
~
is uniformly bounded for
z
E
G, where
1
(J
j, k E IN , j
such that
and
p E pSk(E;F) imply
DEFINITION 2.2:
Let
S be a differentiability type from E to
all functions
&sm(V;F) as the vector subspace of
f
such that, for x
and the mapping We endow
E
U
-+-
akf(x)
x
E
V,
k,
ajp(x) E pSj(E;F) and
x E E
define the space
~
~
k
F.We
&bm(V;F) ~J
0
IS > O}.
such that
p(f - g)
>
for
IS,
g E A, where
p(h)
Consider -
:IN}
U, k .:: rn, k E IN,
a. non-empty -6u.b-6'!..t, then
B(a,k)
T
E
.s.
E
I (a,k,e)
PROOF:
am
m1 k
0, 0
PROPOSITION 2.8:
every
.s.
U, k
I(a,k)
A
A c
E
-1 sup {lid h(a)lIai 0 < 1 < k},
V = {h E ,am(U;F); p(f - h)
neighborhood of
If there exists
< e / 2},
which
is
a
f.
h E V (\ B(a,k)
I
we have
p(h - g) < e/2
for
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
some
g E A. Then:
p (f - g)
B (a,k)
is closed.
DEFINITION 2.9:
+ p (h - g)
c.ond,[t,[on (L),
'[6
'[J.> a.n '[deal 06 (W,G)
J.>a.t'[J.>-
6,[eJ.> c.ondit,[on (L).
PROOF: h
If
h E &cm(U) and
a g) E
(I{)
I{)
g E W, then
oW. Therefore
I{)
aW
hg E Wand, so
I{) 0
(gh)
is an ideal.
Suppose now that (W,G) satisfies (L) and let
g E G and
V c U
be a non-empty open subset such that
g(V) C U. If we consider K C V
and
~
LeE
compact subsets, fEW, k
m,
£
> 0,
there
hEW
is
such that
(x,y) E K
x
L,
0 < i
< k.
Then:
(x,y) E K
This proves that
LEMMA 5.3: 60ft J.>ome
16
(I{)
SuppoJ.>e tha.t
x
L,
oW) a (gIV)
iE
0
< i
to the c.lo).,ufte 06 G
in
tC(E/E),
GeE' ® E, a.nd tha.t (W,G) J.>a.t,[J.>6,[eJ.> c.ond,[t,[on (L). fEW, then
I{)
of
belongJ.> to the Tcm-c.loJ.>uJte
06
I{)
oW
,[n
&cm(U).
.., PROOF:
Consider
fEW, a E U, k < m,
£
> 0
and
LeE
a
compact
GUERREIRO
182
subset. There is
Y
E
L,
0 < i
g E W such that
< k.
Then:
Y E L, 0 .2. i .2. k, which proves that
c.ompfex fLegufaJt i6 and onty
i6 ..Lt6
COMi-6tJ.> only 06 He.fLmite matfLice.-6 and
(2.2.8)
06 matfLice.J.> w-i.th at mO-6t ~wo non-zefLo fLOW-6.
Unfortunately, the known proofs are not simple. If
E is not regular, its "distance from regularity"
measured by its defect, given by formula (2.1.4). Using an
can
be
argument
from [ 5 J we can prove
THEOREM 2.8:
FOfL a Ylo~mal Polya matfLix w-i.th e.xactly p odd -6UppofLte.d
-6equence-6,
(2.3.3)
d
0
60lt wh.ic.h the.Jte
be d.i66eltent ItOW.6 06 E and .6Up-
p
Rl (R ) be the .6et 06 even 2 aILe
(odd)
onu,.in PO.6.i.tiolU (ij,k), j=l, •.. ,p.
Then the 60Uow.ing inequa./'.itie.6 alte nec.e.6.6a1LY 601t the Itegu./'.aJtity 06
(2.4.5)
E
max (
(ii)
I Rl i , I R21)
0) •
U E.)
J
a
c(
6OI!.
wh,[c.h
ll"" ,fp >1) ~ep~e-
-
= AI'" OM
the
j
A c.ctI!Jl..iu
E, then
= y(E l )
E ),
U
OJ!. d ell.
A
-bhi6t6:
ho~~zontal -bubmat~ix
(iiE ) 1
i
J.. i.6 unique, U hM
(~educ.ing)
A 06 Mde~
a(AE
A 06
have no c.oll~~ion.6. The ~h,[6t
by -bimpfe
a{E
-
mult~ple ~hi6t
only one
f l', ... ,f'p . Although
~entation-b
16
A 06
a~e
J
Ej
and
into
(c)
E.
AEi and
~ow~
(b)
muft~pfe ~h~6t
a
Ao.
any
(3.1.8)),
(E \ E ) doe-b not -ba.t,[-b6y the polya. l
mufthe
c.ondi-
n.
For a system
S of differentiable functions and the matrix A(E,X)
associated with
E, X, S, we want to find the partial derivatives
the determinant
D(E,X) of (2.1.3). To differentiate
respect to one of the parameters row of (2.l.3) which contains
D{E,X)
of with
xi' we have to differentiate
each
xi' This leads to
(3.2.2)
(and a similar formula for mixed derivatives), where the sum is taken over all representations of the multiple shifts of order i - th
row of
Xi as
in
Xi approaches another knot
D(E,X) as a function
x j • This behaviour can be de-
termined through the relationships between shifts, collisions, coalescence.
THEOREM 3.3:
Fo~
Xi
~
xj '
D{E,X)
ha..6 the
Taylo~
(Xi - x.) a (3.2.3)
the
E.
We would like to examine the behaviour of of
S
D(E,X)
a!]
(- If C D(E,X)
expa.n-b~on
+ •.. ,
and
206
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
whe.fc.e
C i.6 de6.i..ned in PJtopo.6Ltion 3.2(b), a
a (E.
~
U
E.).
J
and
o i.6 the inteJtchange numbeJt.
For polynomial interpolation, when
S is the system (2.2.1),we
have
THEOREM 3.4:
(ii)
(i)
16
El i.6 a hOJtizontai .6ubmatJtix 06
.i...6 a polynomial in the vaJtiable.6
x.
06 joint degJtee not gJteateJt than
y (E )· l
In a .6ingle vaJtiable D (E,X)
x!
with
coJtJte.6ponding to
El
coJtJte.6 po nding to a Jtow E. in E, 1
x.
~
ha.6 the highe.6t teJtm
~
D(E,X)
(3.2.4)
1
E, then D(E,X)
YT
y = y (E ) and i
(- 1)0* C* D(E*,X*) + •••
3.3. APPLICATIONS OF COALESCENCE
gularity, we take rows
Ei
and
de6ined by the maximal coale.6cence.
E*
For order regularity or order sinE
j
to be adjacent in (3.2.3) whereas
for real or complex regularity this is not necessary. This remarkapplies throughout this section.
3.3.1.
Suppose that
E is a normal Polya matrix. We can give
simple proof [19) of Theorem 2.2. If in (3.2.3), then the same is true of of two rows, we finally reduce
a very
D(E,X) is not identically zero D(E,X). Byrepeatedcoalescences
E to the one row form
{l,l, ... ,l},
for which the determinant is the Vandermonde determinant of the system
S
=
{go, .•. ,gn}. Therefore,
THEOREM 3.5:
16
E i.6 a nOJtmal Polya matJtix and the Vandvtmonde. de-
teJtm.i..na.nt 06 the .6y.6tem S i.6 not identica.lly Zl2.Jto, then not identically zeJto.
D(E,X)
i.6
206
LORENTZ and RIEMENSCHNEIDER
3.3.2.
If the determinant
changes sign, then so does
THEOREM 3.6:
16 one.
on
D(E,X) or
D(E*,X*) in (3.2.3) or (3.2.4)
D(E,X). Hence [13], [19]
E
the. c.oa.£.e..6c.e.d ma.tlt.ic.e..6
E* i.6
olt.
!.ltlt.OYlg£.y
.6iYlgu£.aJt, the.n .60 i.6 the. oJt.igina.e matlt..ix E.
3.3.3.
We can exploit the interchange number
(3.2.3)
(and even in (3.2.4) [22]), by comparing them after coalescence
0
occuring in formulas
of several rows in different ways.
THEOREM 3.7:
Le.t
It.OW.6 F l , ... ,F q 06 E be. c.oale..6eed q - 1 time..6, in two di66elt.ent way.6, to pnoduee. the .6ame. .6ingle. now. 16
0 ,,,, ,Oq_li 1
0i, .. .,o~_l
q.:. 3
a.Jte. the. eoJtJte.6ponding inteJtehange VlWI1be.lt.6,
and i6
(3.3.1)
*
0 1 + ... + 0q_l
0i + ••• + o~_l
(mod 2)
then E i.6 .6tJtongly Jteal .6ingula.Jt. 60.11. any .6Y.6tem S. The. .6ame e.eU.6ion ho£.d.6 60.11. .6tJtong oJtde.Jt .6ingulaJtity i6, in addition, F , ••• ,F l q
the JtOW.6
aJte adjaeent and a.l.e eoale..6eenee.-6 aJte in one diJteetion (Le.
to Jtow
It.OW i
c.on-
i +1, on Jtow
i
to Jtow
+ I
il.
The last statement is required since the coalescence of row to row
i+l contributes the sign from (xi - xi+l)Cl. This
i
contribu-
tion is the same on both sides if all the coalescences have this same direction. In general, a less simple statement, taking into the collisions for coalescences tions of the
q -1
i
to
i + 1, is true when the direc-
coalescences are free (see §6.2).
We give a more explicit formulation of Theorem 3.7 rows of
Ei
Fl
=
account
(R.i'···'JI.~), F2
=
(Jl.l'·"'JI.~) and
mean the posi tions of the ones of row
for
three
F 3 • By (Fl s ' we
F pre-coale:3ced wi th
respect
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
207
to row Fs' and by (F)st - the precoalescence with respect to (F Further, we adopt other
s the convention that two sequences following
mean that their elements should
be written out in the
order. By considering the interchange numbers for
PROPOSITION 3.8:
U
Ft)O. each given
the coalescences
The matltix E i-6 -6tltongty lteat -6ingutalt i6 it con-
tain-6 thltee ltOW-6 60lt which the two -6equence-6
and
Thus, a matrix can have three rows that are so bad that it singular for any arrangement of ones in the other rows, and for systems
3.3.4.
is all
S.
By properly selecting knots
PROPOSITION 3.9:
16 y. -
(3.3.2)
J.
i-6 odd, whelte
Yi
xi' Theorem 3.3 and 3.4 give us
= y(E i
06 cotti-6ion 60lt ltOW i
)
in
polynomial inteltpotation,
l: Il j~i ij
and
Il ij
E, then
= Il(E i E
U Ejl alte the coe66icient-6
i-6 -6tltongty lteat -6ingutalt 60lt
208
3.3.5.
LORENTZ and RIEMENSCHNEIDER
We have restricted ourselves in (3.2.3) to the expansion
D(E,X) in one variable for sake of simplicity. If several knots
of xi
approach
x., we obtain multiple coalescence. The expansion will then J contain, as its main term, a form of order a in several variables. If this form changes sign, the matrix
E must be strongly singular. This
requirement is particularly meaningful for real singularity when the values of the variables of the form are unrestricted.
3.4. EXAMPLES: trices
E and
Let
E be obtained from
E by coalescence.
The
E can be regular, weakly singular or strongly singular
in logically nine possible combinations. Theorem 3.6 rules combina tions:
ma-
E strongly singular wi th
E being regular
out or
two
weakly
singular. All the other combinations can occur. Indeed I by coalescing the matrices
E
l
, E2
and
E may be regular when examples of
E3 of (2.4.1) to two row form, we see that
E is any of the three types.
E. Kimchi and N. Richter-Dyn [16]
The
following
are less trivial.
o o o 1 o 1 o o o o o o o o
1
1
o o
o
1
o
1
o o o
o
1
100
0
o
0
1
0
1
o 1
0
0
1
0
0
1
1
o
I
0
0
0
o
0
I
000
0
0
00101
0
0
0
11000
0
1
(3.4.1)
E
5
0
0
=
The matrices
0
E and
lar; and the matrix
ES are weakly singular; the matrix E4 E6 is strongly singular.
is regu-
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
209
§4. INDEPENDENT KNOTS The connection between the concept of an odd supported sequence and the extended form of Rolle's Theorem was exploited in
§ 2.3
to
obtain a simple proof of the Atkinson-Sharma theorem. This simple connection suggests that a more detailed study of the information gained from Rolle's theorem is warranted. The method of independent was formulated by Lorentz and Zeller [28] and developed
knots
further
Lorentz ([18], [20]), in order to study singular interpolation
by ma-
trices. Let E be an
mx n +1
differentiable function on
x
=
(xl' ... ' x ) m
C
interpolation matrix, and f be an n-ti.rres [a,b)
which
E
and
and its derivatives specified by (4.1), we
can
[a,b 1, that is, let
(4.1)
f(k)
From the zeros of
f
is annihilated
f
(x.)
satisfy
1
1
by
in
E.
derive further zeros by means of Rolle's theorem. A selection
of
a
complete set of such zeros is called a "Rolle set" of zeros. A Rolle. .6e.t
R 601t a 6u.n.c.t.ion.
f
annihilated by
E, X is a col-
, k = 0,1, ... ,n, of Ro.e.le. .6 e.t.6 06 z e.ltO.6 (with mul tiplicik ties specified) 601t e.ac.h 06 the. de.lt.ivat.ive..6 f(k) selected inductively lection
R
as follows: The set (4.1). If
Ro
consists of the zeros of
f as specified
in
RO, ... ,R k have already been selected, then we select Rk + l
according to the following rules: 19
A zero of f(k) in also a zero of
29
All zeros of
Rk of multiplicity greater than one
f(k+l) with its multiplicity reduced by one. f(k+l)
(including multiplicities) as slJ8cified
by (4.1) are included in 39
is
For any adjacent zeros
ct
Rk + l • I S of
select, if possible, a zero of
f(k) belonging to R , we k f(k+l) between them subject
210
LORENTZ and RIEMENSCHNEIDER
to the restrictions: (a) If the new zero is one of the
xi' then it is not liste1in
(4.1), or (b) there is an additional multiplicity of f(k+l)
zero of
t
is the multiulicity of
is defined as follows.
=
f(k+t) (xi)
as a zero of
xi
as a
We add to (4.1)
the~
0, and determine the multiplicity of xi
f (k+l) from these equations. This may connect
two sequences in than
of
f(k+l) given by (4.1), then the multiplicity of xi
Rk + l
tion
as a zero
which is not acknowledged by (4.1).
(c) In the event of (b), if
in
Xi
E and prescribe
a
raul tiplici ty
larger
t + 1.
If a zero does not exist subject to the restrictions in 39, then we say that a
lo~~
occurs at step
k + 1. A Rolle
set
constructed
without losses in any of its steps is called maximal. The function f may have many Rolle sets, some of them may be maximal, while
others
are not maximal. Some properties of Rolle sets are immediate consequences of the selection procedure. First of all, the only multiple zeros of f(k+l) in
R + k l
are among the points
xi
in
X. Secondly, the extended fom
of Rolle's theorem shows that a loss will not occur if the rows of E corresponding to
xi between adjacent zeros of
ported sequences.
LEMMA 4.1:
Rolle
~et~
Rk contain no odd sup-
(This was the connection used in §2.3.) We have
16 -the
mabtix E
06 a 6unction
f,
ha~
no odd 6uppolt-ted
annihila-ted by
~equenc.e~,
E, x, a.lte
t:henaU
ma.ximal.
The number of Rolle zeros in a maximal Rolle set can be determined by induction:
LEMMA 4.2:
16
f
i~
annihilated by E,X, then 6011. eac.h k, k=O,l, ••• ,n,
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
the numben 06 Rotte zeno~ 06 i~
6uncUon f S
Let
=
at
tea~t
211
in a maximat Rotte ~et
f(k)
be a system of n-times continuously dif-
ferentiable functions which are linearly independent [a,b]. A set of knots
with respect to the system every polynomial
P
in
the
M(k) - k.
{go, ••• ,gn}
subinterval of
60n
S
[a,b]
XC
on
each
is called independent
S if for each interpolation
annihilated by
open
matrix
E,
has a maximal Rolle set
E,X
of zeros. Using a weak form of Markov's inequality, which is
valid
each system S, it is possible to show that Rolle zeros for can be selected away from the zeros of
for
p(k+l)
p(k). More precisely,
(see
[37) for algebraic polynomials)
LEMMA 4.3:
~uc.h that i6 60n
~ome
i~t~
Thene
i~
i3 - a
inMea~ing
a monotone
2. R., a .::. a
6unc..t.i.on 6(,I',),O'::'6(R.) < %R,
< i3 .::. band
potynomiat p in Sand
p
=
(k) (a)
p
=
(k) (13)
k, k = O, ••. ,n -1, then thene
~, a + c(R.) .::. ~ < 13 - cU,) OM which
0
ex-
p(k+l) (~) = o.
For simplicity and without loss of generality, we take [a,b] = [-1,1.1. From Lemma 4.3, one derives
p, 0 < p < 1, thene i~ a ~equ.enc.e
THEOREM 4.4:
Fon eac.h
with
having the 6ottowing pnopenty. Let
p < Yl
[-p,p] u {± y }, and s ~upponted ~equenc.e~
Then each potynomiat RoUe
E
be
~u.b~et
a
be an intenpotaLion matnix which ha~
in the p
X
{±YS};=l
now~
c.onne~ponding
in S, annihitated by
to
knot~
E,
X,
06
no odd
Xi' - p,::,xi,::,p. ha~
a
maxima!
~et.
For the proof, the points
ys
are chosen inductively very close
to 1 so that the selection of Rolle zeros in step 39 is always sible. It is essential for the proof of Theorem S.l - indeed, the main idea - that the "harmless" knots
Xi
posit is
can be made variable in
212
LORENTZ and RIEMENSCHNEIOER
an interval
(- p, p), arbitrarily close to (- I, 1). Clearly, any knot
set X contained in
{±
Ys}
is independent with respect to the sys-
tem S. Theorem 4.4 gives another simple proof of Theorem 2.2 (Windhauer [47] or [20]). Assuming that
E is a normal Polya matriX,
we
take
X c {± y } and show that the pair E, X is regular. Indeed, a polys nomial Pn annihilated by E, X is identically zero by a standard ap-
plication of Lemma 4.2. As has been pointed out in [19], Theorems 2.2 and 2.4 extend to equations of the form
(4.2)
where
1) ,
D.
J
are certain differential operators of order I, and S is the
Chebyshev system connected with these operators (for a definition of S, see [15, p. 9, p. 378- 379]).
§5. CLASSES OF SINGULAR MATRICES
The Atkinson-Sharma theorem provides only a sufficient
condi-
tion for the regularity of matrices; the condition is not necessary. However, a good guiding idea is that. this condition
is
"normally"
necessary, or at least necessary under some simple additional conditions. All theorems of this section refer to
inte~poiation
by
aige-
b4aic poiynomiai6 and 04de4 6inguia4ity.
THEOREM 5.1:
An
mx (n+l) nMmai Bid.h066 mat4ix i-l> -I>tMngiy 6ingu.-
la4 ' 0
is a constant.
THEOREM 5.5 [25]:
Fo~
each
E > 0,
the~e ~~
an
nO
216
LORENTZ and RIEMENSCHNEIDER
60llow~ng
(5.4) but
w~th
n > nO ' at m04t
eB(m,n)
00 them have
THEOREM 5.6 [25]: 60~
m04t
Among all
p~ope~ty.
n ~ no' eP(m,n)
a~e
eB(m,n)
B~~kh066
mat~~~e4
a~e ~egula~.
What
~4
mo~e,
all
4uppo~ted 4~ngleton4.
Fo~ ea~h
among all
B(m,n)
e > 0, the~e ~4 an
P(m,n)
Polya
mat~~ce4
nO
= note)
40 that
4ati40ying (5.4),
at
4egula4.
§6. THREE ROW MATRICES
6.1. ALMOST HERMITIAN MATRICES
It is not clear in what respect the
theory of regularity becomes simpler for three row matrices. The theorems on coalescence are not strong enough to reduce the general case to this one. Furthermore, we shall see that even very simple three row matrices present considerable difficulties. The results of
§6 refer
to order regularity. We shall study
3 x (n + 1)
normal Birkhoff matrices with the
following placement of ones
elk
I, 0 < k < p; e 3k
=1,
0 < k < q;
(6.1.1) 1.
Then
p + q + 1 = nand
also assume that
k2 < n; without loss of generality, we
shall
kl < k2 - 1, P .5. q. For the knot set, we shall take
X={-l,x,l}. One of the smallest matrices of type (6.1.1), E3 of (2.4.1),has served to show that regular matrices can have odd supported
~aes.
Generalizing this example, several authors (DeVore, Heir and
Sharma
[6 ] , Lorentz and Zeller in [19), and Lorentz, Stangler, and
Zeller
217
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
[26
J)
studied matrices of the form (6.1.1). It was hoped that in this
way the problem of regularity could be completely solved for at least one nontrivial case. The incomplete success of this attempt leads one to believe that i t is hardly possible to express the property of regularity in terms of simple properties of elements The method of the paper [6 classical Jacobi polynomials
1
e
ik
of a matrix E.
was to apply known facts about the
pea,S) (x). In [19], n
the
alternation
properties of zeros of derivatives of the polynomial (1 + x) p (1 - x) q were used. The first method gives more detailed information while the second method is applicable to wider classes of matrices.
THEOREM 6.1 [26 nec.e~~aJty
J:
In oJtde.Jt that the. matlL-tx (6.1.1) be. Jte.g ulaJt , li.u.,
that
p + q + 1,
(6.1. 2)
k2 > q
(6.1. 3)
In the
c.a~e.
(6.1.2),
E -t~ JtegutaJt
-t6 and onty -t6
(6.1.3), the. ma.tJt-tx c.an be. eitheJt JtegulalL oJt taJt-tty
06 .the. ma.tJtix
tlLix w-tth
paJtame.te.Jt~
p = q. In the. c.a~e.
~ingutaJt,
(6.1.1) -tmptie~ .the lLegulaJtity -tn~tead
ki' k2
06
k l , k2
but the Jtegu-
06 a
~imilaJt
ma-
-t6
OJt
{Note that inequalities (al have been stated incorrectly in the paper [26
J, namely with
ki
~
kl < k2
~
ki. This error occured in the
last lines of the proof in [26, p. 435J. The inequality (5.5) should
218
LORENTZ and RIEMENSCHNEIDER
be replaced by the reverse one:
"(5.5)
YI+1(A) 2. yiP,) for some 1.")
The proof of this theorem is by the "chase method". As a didate for the nontrivial polynomial annihilated by
can-
E, X, we take
P(x,A)
We let
A change continuously from - '"
to those zeros of
P
(k ) l
and
P
(k ) 2
to +
00
and study what happens
whose existence is guaranteed by
Rolle's theorem. The matrix is singular exactly when one of these zeros overtakes the other at some
xO' for then
P
(k ) l
(xO,A)
=p
(k ) 2
(xO,A) =0.
The second part of Theorem 6.1 means that in the triangle given by
P
y =
A(X),
~
x, Y
~
q, x + 2
~
y, there exists a monotone increasing
with slope at most one, so that
on the curve, and regular below it. For
is singular above
E
p
= 1,
~
and
this curve was
dis-
covered in [6 I, and was shown to be the upper branch of the ellipse
(6.1. 4)
(q
+
2)
(x + y -
1) 2 -
4 (q
+
1)
xy
0;
moreover, E is weakly singular on this curve and strongly
singular
above. For some values of the parameters, the statements
of
6.1 were proved also in (6 I and [19]; in addition, it was
Theorem possible
to distinguish between strong and weak singularity. One general case of weak singularity has been found to date, namely when
q = p + 1,
kl + k2 = P + q + 1. For more details, consult the paper of
DeVore,
Meir and Sharma [6 I.
6.2. CRITERIA BASED ON COALESCENCE Polya matrix. For the knot set
X
Let
E
be a
3 x (n + 1)
{O,x,l}, the determinant
is a polynomial in x. Clearly, E will be strongly singular sign of
D(E,X) is different in (0,£) and (n,l) (e:, l-n
normal D(E,X) .if
the
sufficientlysmall~
RECENT PROGRESS IN BIRKHOFF INTERPOL.ATION
219
This simple observation is the essence of several criteria strong order singularity of
E,
although the statements
for
the
themselves
appear totally unrelated. There are several equivalent forms in which this comparison of signs can be carried out. One of them is given by the special caseof Proposition 3.8 when the matrix E consists just of the three ordered rows
F I' F 2' F 3 (of course, the interest of Proposition 3.8
is not
limited to this case). Another form is one given by Karlin and Karon (.t ,.t , ... ,.t ) = F2 be the positions of the q l 2 1 1 3 3 (.tl, ... ,.t ) and (.tl, .•. ,.t ) be their posiq q
([13, Theorem 2.3): Let ones in row 2, and let
tions in the pre-coalescence of row 2 with respect to row 1 and
row
3 respectively.
PROPOSITION 6.2 (Karlin and Karon) : ma..tJr.).x, :then
E
16
E
)..6 a. 3 x (n + 1) nOJtma..t po.tya.
)..6 .6.tJr.ong.ty .6).ngu.tcOt when
.t~-l
.t~-l (6.2.1)
q
~
j=l
{
J~
M(.t. -1) +.t. +.t.1 +.t.3 + J~ J J J J k=l. J
In 6a.c.t, :th-i..6 l>um need only be :ta.f<en oveJt
} e3 k
k=.t. J j
:the 6-i.Jt.6t e.temen:t.6 06 odd l>uppoJtted .6eque.nc.el>
OM
06
::: 1 (ood 2) •
'
wh-i.c.h :the.t j
a.Jte.
Jtow 2.
The method of Karlin and Karon was to analyze the signs of the determinants involved by using arguments from the
theory
positivity due to S. Karlin. For the last statement of the one verifies that adjacent ones, contribute
0 mod
2
.tj+l
of
total
theorem,
.t. + I, or unsupported ones, J
in (6.2.1).
Both Proposition 6.2 and Proposition 3.8 are consequences
of
coalescence and the use of the Taylor's formula (3.2.3). This can be. explained best of all if we define "directed coalescence" as follows. If
F , F2 are two adjacent rows of E, we define the directed l alescence F 1 => F 2 as the matrix derived from E by replacing
corow
LORENTZ and RIEMENSCHNEIDER
220
F1 by its pre-coalescence,
°1 ,2'
number,
Pl ,
with respect to
of the coalescence
F1
~
F2
F 2 . The
interchan~e
is the number of
inter-
changes needed to bring the sequence of integers
F1 , F2 into natural order. (Here a row F is represented by the positions of the ones as in § 3.) In a similar way,
F1 .. F 2
replaces row F 2 by its pre - co-
alescence wi th respect to F1 and has the interchange number Then (by (3.2. 3)
° 2,1 .
)
(6.2.2)
where
,2 = u(F U F ) is the coefficient of collision. For calcu2 1 l lating the interchange numbers of further directed coalescences, the u
positions of the ones in
F1 " F2
and
F2
F1
=>
are assumed to bein
their natural order; then, for example, cr 3, (2, 1) = cr 3,( 1, 2 ) directed coalescences F3 ~ (F .. F ) and F3 ~ (F l ~ F ) l 2 2 tive1y. Let
Ci
for the respec-
be the sum of the exponents of powers of (- 1) giving the
signs mentioned at the beginning of this section. We can give several equivalent expressions for
Ci
(mod 2) by means of directed
coa1es -
cences and Theorem 3.3. For example, to obtain Proposition 3.8, estimate the sign of
O(E,X)
near
0 by means
F ) * F , and near 1 by means of 2 3 this way (F l
~
(6.2.3)
Fl
=>
of
the
we
coalescences
(F 2 * F ) andobtainin 3
o - 01,2 + cr (1,2),3 + 02,3 + 01, (2,3) (mod 2).
To obtain Proposition 6.2, we consider the coalescences (Fl " F2) .. F3 and F " (F => F ); this gives 2 l 3 (6.2.4)
221
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
(to see the equivalence of (6.2.3) and (6.2.4) directly, (6.2.2) and an extended form of (3.1.7». 8
=1
Equations
one
uses
(6.2.1)
(mod 2) can be shown to be equivalent by the careful
and
computa-
tion of the collision and interchange numbers of (6.2.4) by means of the quantities in (6.2.1). Similar ideas give a special case of Propositions 3.8 and found by Sharma and Tzimbalario (42). Let Birkhoff matrix with ones in positions and let
Fl
E be a
3 x (n + 1)
=(ll,···,t~),
6.2
normal
F3=(tl', ••• ,l~'),
be the positions of the zeros in row 2.
PROPOSITION 6.3:
16
kl > max(l~ - p, C' r
r)
a.nd i6
p ~
(6.2.5)
j=l
.then
E
(k r + , - kJ') + pr - 1 (mod 2), J
i-6 -6.tJtongly .6inguta.lL.
Here we use the coalescences (F 1 ~ F 2) ... F 3
and F 1 "* (F 2
0
mean
ur
if
u < 0, except that it is not defined if both the kernel
u
u .:: 0, and
= 0,
r
= O.
We
o
if
obtain
D(E,X) of (2.2.2) by ren-k-l placing the elements of the first col\.llU'l in (2.2.2) by (xi-t) + /(n-k-l)!:
(7.1.1)
If
KE(X,t) from the determinant
K(t) = {
n-k-l (xi - t) + (n - k - l)! '
n-k-l -k xi xi (n - k -1) ! ' •.. , (-k)!
Dik(X) are the algebraic components of the first column elements
of the determinant (2.2.2), defined for
.=
1, then
n-k-l ( xi _ t) +
(7.1. 2)
(n - k - 1) !
If the knots are ordered, xl < '" is a polynomial in
e ik
t
K(t) m then the determinant in each of intervals (-00, Xl) , (~,x2) , .•. ,(xm' +(0) ,
hence a spline. One sees that
< x '
K(t) is zero outside of [xl,xml. {The
same applies to the derivatives of the ken'lel. '!hus, K(j) (t), j =0, •.. ,n-l, [A j ,B j J , where A. (or B j ) is the smallest (or ] } = 1 for some k ~ n - j - 1 the largest) of the x.~ with e ik Integrating (7.1.2), we obtain
is zero outside of
.
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
(7.1. 3)
Let
223
D(E,X) .
An denote the class of all (n - 1) -times continuously dif-
ferentiable functions
f
on
[a, b) for which
f (n-l) is
absolutely
continuous.
THEOREM 7.1 (Birkhoff's Identity [3) ): ZeJWf., in. .the laf.,.t column..
FOIL ea.ch
f
E
Le.t
E be a. YWItmcU'. mcU'lUx w.uh
An an.d each f.,e.t 06 k.n.o.tf., X in.
[a,b) ,
tf(n) (t)K(t)dt.
(7.1. 4)
a The simplest special case of (7.1.4) is Taylor's formula
with
integral remainder. From this theorem, we can obtain mean f E Cn[a,b). If
value
formulas.
Let
K(t) does not change sign, then by using (7.1.4) and
(7.1.3), we can obtain
The same is true if
K is of arbitrary sign, but
of degree not exceeding e
ik
n.
f
is a
polynomial
In both cases, the relations f(k)(x.) = 0, ~
imply fen) (~) =0 for some ~, xl < ~ < x ' m Suppose now that E is a normal Polya matrix without any
= 1 and D(E,X)
strictions and
to
X is a set of knots for which D(E,X)
there exists a polynomial
p(k) (x.)
n
~
P n of degree at most
f(k) (x.) , ~
1.
F
re-
O. If fE An+l'
n for which
224
LORENTZ and RIEMENSCHNEIDER
We would like to get a formula for the difference extend
E by adding a O-th row with only a single one I e OO
E to
and by adding an (n + 1) -st column of zeros. Let the
xi
kernel
then
I
f(x) - Pn(x).
=1
I
x be different fran
X is the set of knots obtained by adding
= KE(X,t)
K(t)
We
x to
X. The
is the Peano ke~nel of the interpolation.
One ha~ 60~
THEOREM 7.2 (Birkhoff):
1
(7.1. 6)
D(E,X)
X C (a,b)
Jb f (n+l) (t) K (t) dt a
{A similar formula holds for the difference f(k} (x)
- p!k} (x)
if we insert the one in the new O-th row in position k, Le. e
O,k
I
=l.}
7.2. NUMBER OF ZEROS OF SPLINES
The deepest theorem of Birkhoff in
[ 3 I counts the number of
06
chang e~
I.>-ig n of a kernel
KE
estimate is also valid for other splines (D. Ferguson [8 eralization by Lorentz [21] concerns the numbe~ A function
S
on (-
00 ,
if there are points gree
~
< xm
Xl
p.Une 06
= (e ik )
= 1, ••• ,m,
at
x .• J.
I6
Let
be an
and that
S
m x (n + 1)
e
ik
=1
P -il.> the numbeIL
-i6 the numbe.IL 06 one.-6 -in the-iJr. mult-ipli.c-it-iel.»
(7.2.1)
gen-
of splines.
S
is a polynomial of den
for
at
i , and is zero outside of (xl,X ). Let [a,b] be the smallest m S vanishes.
i
ze~O-6
A
+ (0) is a spline of finite support of degree n
interval outside of which
E
1).
n on each interval (xi,x + ), has exactly degree i l
least one
Let
06
(X, t). This
06
matIL-ix
deg~ee
06
wheneveIL
06
n -1 w.i.;th "nato Xl < ••. < xm•
ze~ol.>
and onu
1.>0
s(j), j =n -k -1, hal.> a jump
odd ~uppolL.ted 6e.que.nce.6 06
E, and
Z -il.> the
S -in (a,b), then
Z 0,
a < x < b,
j = l, ••. , p,
are given signs, and
given integers. In analogy with the case
< n
kp
kl
= 1,
are
this is still
called the problem of monotone approximation. Even more generally, one can restrict the ranges of the derivatives
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
227
by [38] (k .)
(8.1. 2)
L(X)'::'P ]
n
]
For the bounding functions u.
=+
and that ei ther
R,j
that either
]
(x)
R.
o
aR.M.6 on peut tnouven une 6onctlon
q p0.61tlve. ven161ant
x
242
MALLIAVIN
2.1. 2.
q (x),
PREUVE:
L' hypothese de convexi te fai te sur
q(x)
2.1. 3
x m(t)..£t t fo
ou
x
q
m(t)
> O.
implique que
est croissante.
2.1.1. implique que
f Soit
_dm(tl < tl/ 2
co
•
Q(z) la transformee de Mellin de
q.
Transformons
for-
mellement les deux membres de 2.1.2. par Mellin, remarquant que
cotg 11 z z
- 1 < Rez < 0
on obtient
o(z) 2.1.3. donne, notant par
z tg ( 1TZ) Q ( z) .
M la transformee de Mellin-Stieljes de
Q(z)
M(z)
---;r-
d
Notons par l1Z
l1Z
h(x) la fonction ayant -
~
< Rez
- log p(x)
C
X E
I
1.1.
E •
t
11 resulte du fait que cette integrale est> -
00
que
Io djJ=jJ(t)
est
une fonction continue. Soit n(t)
partie entiere de
jJ(t)
et soit exp [-
I
log(l - zt-l)dn(t)] = F(z).
F(z) est une fonction meromorphe n'admettant que des poles simples. D'autre part, posons
s (t)
3.1. 3.
I
o
jJ (t) - n(t)
log 11- zt -11 ds(t) =Re
I
< s (t)
< 1
t _z z S(t)dtt = ReIooS(XU) l+iT u-l-iT du U
o
246
OU
MALLIAVIN
T = yx- l ; soit
a
alx +
fo
tel que
J1 / 2
+
a/x
< I, a > 0
pta)
+
J2 1/2
J+ 2
OO •
La premiere integrale est inferieure
Cll(t)x~t La seconde
a
La derniere Reste s
=
a
0(1) .
t
log x + 0(1).
a
0(1).
evaluer la 3eme integrale
on
Ie
fera
en
posant
! s11l~I:r' d' ~ ou
:r
sl + 1
Re
dt
a
2 s(xu) 1 l + 'iT ~ d 1 < -2 u- -n u
J1;'2
+ / Re
f
1 + i T u-l-iT
f112 2 IRe
1 + i T jdu +-' 1 J2 Re~ l' dU/ u-l-i T 2 I 1/2 u-l-n
~)
(1 -
du / .
La premiere integrale < - log T + 0 (1), la seconde et la tro:l..sierne sont 0(1), d'ou en tenant compte de 3.1.1.
xA !F(X + iy)! < Bp(x)y,
x
E
E,
!y!
A + 2, b l , .. " b ' r pOints de E distincts; r alors on peut trouver une fraction rat:l..onnelle H(z) ayant les bk Soi t
r
pour poles simples et telle que
H(z)
o(z
-r
),z+co
F(z) H(z)
verifiera
APPROXIMATION POL YNOMIALE PONO~R~E ET PROOUITS CANONlaUES
3.1. 3.
IF1(Z)I a> 0,
APPROXIMATION POLYNOMIALE PONoilRilE ET PROOUITS CANONIQUES
on peut dans 4.3.3, prendre
~lors
Ie produi t canonique construi t
PREUVE:
avec
ds
= 1;
de plus 4.3.2 a lieu si
est simplement posi tif.
Posons
nIx)
Alors
a (x)
251
dn
s (x)
r
PI (t) a(t)s(t)
J E,,[x;+oo]
dt t
a pour support E, et
dp
PI (t) a(t)s(t)
d(...!!.)
s
dt t
tEE
o Escrivons 2.1, remarquant
'I-
t
F(x,t)dp(t) +
pIx)
a
E •
0(1) et utilisant 4.3.2,
(Pl(x».
On a si
F(x,t)
ou
f
s(xS;) S; - 1
..9.£
< 8s(x) logll -
~I
S;
8 > 0, et une evaluation analogue pour
t
E
3
[x'2 xl
d'ou
J F(x,t)dp(t)
ou
8
1
Pre nons
et
8
n
l
sont deux constantes numeriques positives d'ou
2
8
-1
3
n
et posons
J logll
-
~
I dnl(t) + A
262
MALLIAVIN
ou - A
sup [ PI (x) -
tp
(x)
f,
0 < x < Xo
Le resultat suivant classique pour les fonctions entieres d'ordre
~ s'etend a 2
tp(z):
i l existe une suite infinie de cercles
tels que
e.
uniformement en
Dans
{iz i
tiere, donc
tp(z)
< ~} () CE, tp(z) est harmonique negative surla fron-
z.
quel que soit
< 0
5. Nous allons donner dans ce paragraphe des conditions pour que la suite
{xnp(x)}
suffisantes
soit non totale dans l'espace
CotE) des
fonctions continues sur E nulles a I ' infini. Etant donne x
E
I x'
Ix
C
E.
x
E
soi t
Ix Ie plus grand intervalle tel
inf { 1,
J!t t
Lee natatianc etant eellec de 4.3, cuppaconc
lee hypothecec de 4.3, cont catic6aitec
pou~
Pl(x)
= log
pluc cuppoconc ou bien que
5.1.1.
lim inf(- PI (x)
~ a. (x) ) > 0
au bien que log a.* (x) = 0 (PI (x))
5.1. 2.
que
Posons
a. * (x)
5.1. PROPOSITION:
E
- PI (x) a.*(x) (I-log a.*(x)) sex)
et que
log i 1 + vi
I dr (t) > rex> log 11 + 8 ( Jo
donne
~ ) I dr (t) ,
d'ou 6.2.
BIBLIOGRAPHIE
[1 I
G. COULOMB-COURTADE, These, Paris 1976.
[2 I
Y. KATZNELSON, Comptes Rendus 246 (1958), p. 281.
[3 I
P. MALLIAVIN et S. MANDELBROJT, Sur l' equi valence de deux problemes de la theorie constructive Sci.
[4
~co1e
Norm. Sup,
(3)
des
fonctions,
Ann.
75(1958), p. 49 - 56.
I s . MANDELBROJT, GeneJLal :theOJteJM 06 Clo"'uJLe, Rice Institute Pamphlet. Special issue (1951).
[5 I
L. NACHBIN, Element", 06 ApPJtox~mat~on TheoJty, D. van
Nostrand
Co., Inc, 1967. Reprinted by R. Krieger Co " Inc. 1976.
Approximation Theory and Funational Analysis J.B. FroUa (ed.)
©North-Holland Publishing Company, 1979
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
REINHOLD MEISE Mathematisches Institut
der
Universit~t
D - 4000 DUsseldorf, Universitatsstr. 1 Bundesrepublik Deutschland
PREFACE
When the author started to deal with the subject
of
the
present article, he did not know too much of the various ways how one can do calculus in (real) topological vector sIJaces. He was rrainly interested in a generalization of the theory of distributions to infinite dimensional locally convex (l.c) spaces, by means
of
duality
theory and with relations to infinite dimensional holomorphy. Therefore he began studying spaces of differentiable
functions on
1. c.
spaces by analyzing the definitions of Cn-functions used by Aron (3J, Bombal Gordon and Gonzalez Llavona (10 J and Yamamuro [24 J • Then he (re-) invented the notion of
n
tions on an open subset arbi trary covering of
times continuously y-differentiable funcrl
of a 1.c. space
E
(2.4), where
y is
an
E by bounded subsets. As he realized later, this
notion had been introduced with a slightly different definition
by
Keller [18) already. Further investigations showed that many results known to be true on open subsets of
:rn.N carryover at least to Frechet
spaces or strong duals of Frechet-Montel spaces i f one takes as y the system of all compact subsets of
E. In order to be precise, we will
now sketch the main results of the article. In the first section most of the definitions are stated as well as some results which will be used in the sequel. In the second part 263
264
MEISE
we introduce the 1. c. space en W, F) of
n
y
ferentiable functions on an open subset 11 values in the 1. c. space y
F, where
n
times continuously
y -dif-
of the 1. c. space
E with
is a natural number
is a system as introduced above. For open subsets
complete l.c. space and only if
E~o
E,
we show that
COO W) co
11
or
and
00
in a quasi-
is a Schwartz space if
is a Schwartz space.
In the third section we give
a
sufficient condition
for
the
approximation property of Cn (\l). The proof of the corresponding theco orem 3.5 is a generalization of the proof given by Bombal Gordon and Gonzalez Llavona [10] inthe case of Banach spaces. (Actually an analysis of [10] ledto the result presented here.) Since the proof
uses
a criterion for the a.p. due to Schwartz [22], we first characterize n Cn (ll) E F as a topological subspace of C (Il,F). The main lemma (3.3) y y for theorem 3.5 goes back to [10] as well as to Prolla and Guerreiro [20]
(in the case of Banach spaces). It has also some further appli-
cations which generalize parts of the results of [20]
and which may
be of interest in connection with the theorem of Paley-Wiener - Schwartz
C~o(E,~) '.
for the elements of
In the last section we prove that, for open subsets in certain 1. c. spaces
El and
E2
III and 112
respectively, there exists
a
na-
tural topological isomorphism between C~O(1l1,C~oW2» and C~o(\ll x 11 2 ). By the results on the a.p. of tation of
C~o(1l1
x
11 2 ) as
While finishing
his
C~o(m this also implies
(-tensor product
a
represen-
C~o(1l1);( C~o(1l2)'
investigations, the author received
the
preprint [12] of Colombeau, where spaces of COO-functions on Schwartz bornological vector spaces are studied. Colombeau has pointed out to the author that any function in C~o(Il,F) is a Cn-function in Silva's large sense if the quasi-complete space
E is given the compact bor-
nology. For Frechet spaces and for strong duals of Frechet - Schwartz spaces
E
both notions coincide. This shows that
the
bornological
setting is more general as far as the Schwartz property of C~o(ll) is
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
265
concerned. However, it is not known how to prove the results of this paper with bornological methods if
E is the strong dual of a Frechet-
Montel space only. Let us come back to distributions again: It should be remarked that obviously some results in this article can also be regarded propositions on the dual of
as
c:o(n), a space which is a natural gen-
eralization of the distributions with compact support. mentioned in remark 4.8 will be the subject matter of
The a
results
subsequent
paper.
ACKNOWLEDGEMENT:
J. F.
The author thanks R. Aron, K.-D. Bierstedt,
Colombeau, H. Jarchow, H. H. Keller and L. Nachbin for some
helpful
discussions or correspondence on the subject of the article.
He also
gratefully acknowledges partial financial support from GFD and IMU.
1. PRELIMINARIES In this section we shall fix the notation, recall some definitions and state some results which will be applied later.
We
shall
use the theory of locally convex (l.c) spaces as it is presented e.g. in the books of Horvath [17], Kothe [19] and Schaefer [21] . Throughout this article, a l. c. space always means a
Iteat Haus-
dorff l.c. space, because we only want to deal with real differentiable functions.
1.
Let
E and
subsets of
F be 1. c. spaces and let
E which cover
tinuous linear maps from
y be a system
E. Then, on the space E into
bounded
L(E,F) of all con-
F one can introduce the correspond-
ing y-topology of uniform convergence on the sets in ing l.c. space is denoted by
of
y; the result-
L y (E,F). It is well known that this ta.
pology does not change if the system is enlarged in such a way
that
MEISE
266
it is directed under inclusion and that any subset of a set in y belongs to
y. We shall always assl.Ulle that
y has these properties.
finite Yb we denote the systems of all dimensional bounded, all compact, all precompact and all boundeds~ By
Ya' y co ' y c
and
sets of E. The corresponding spaces
Ly (E,F) are denoted by La (E,F) ,
Lb(E,F). We write and
2.
Ly(E) instead of
DEFINITION:
a)
By
E and
F be 1. c. spaces, y a system of bounded
n
IN.
E
£(En,F) we denote the linear space of all n-linearmap-
pings from b)
·E n
into
F.
n > 2; u E £(En,F) is called y-hypocon~inuou~,
Assume for any
k with
1
~
hood W of zero in in E such that 0)
~
k
n, any
{u
£ (En ,F) for y
E
£(En,~)
S
E
Iu
is
neighbour-
n = 1 by
zero
L (E,F) and for
case
sn
for
£y(En,F) with the topologyof ~-
y, we can endow
of
y-hypocontinuous}.
u E £y(En,F) is bounded on
form convergence on the system continuous
£a(En,F)'£CO(En,F), ••• if
yn
linear
{sn I s E y}. mappings
As
we
in
write
Y = Ya' Yco ' •••• The elements
£cr (En ,F) are called ~epalLaA;e.ty con~inuou4 n-.tinealL map-
pi.ng~.
d)
Y and any
u(Sk-l x V x Sn-k) c W.
Since, obviously, any
of
E
if
by
n > 2
the
S
F there is a neighbourhood V of
We define the space
any
instead of
Y
Ly (E,E).
Let E and
sets which covers
E'
We write
£ (En) instead of y
u E £(En,F) is called 4ymme~lLi.c, if for any permutation of
n elements and any
x = (xl' ••• , Xn) E En
we
~
have
SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
U(X'lT (I}""
267
,x'lT (n))
t~(En,F} of
The closed linear subspace
ty(En,F}
is
en-
dowed with the induced topology. The proof of the following lemma is an easy exercise.
3. LEMMA:
a)
Fon any Fo~ any
u E ty(En,F} th~ 6ollow~ng hold¢ tnue:
S E Y the ne¢tn~et~on
u
I sn
~~ un~6o~mly eon -
tinuou¢. b)
any k. w~th
FM ~4
1 ~ k
-
E
n
of
n e
m
C~o W) is nuclear.
But this result is essentially finite dimensional: For let
on
denote the canonical embedding. Then
is an open subset of
mn, hence
COO (rl ) is nuclear for any n co n
Em.
Now it is a consequence of Yamamuro [24 L(1.6.1) ,that COO (m =proj COO (Sl ). co +-n co n Since the projective limit of nuclear spaces is nuclear, this proves the nuc1earity of
C~o(n).
280
MEISE
3. THE ROLE OF THE APPROXIMATION PROPERTY The aim of this section is to derive a
condition for
suffici~
e~o (0). This will be done by an application of theorem
the a.p. of
1.7. Therefore, we first give (under appropriate hypotheses) a charen (0) and a quasi-carplete l.c.space.
acterization of the E-product of
y
Le.:(: E and F bel. c.. .6 pac.e.6, let y be a .6y.6tem 06 bounded
1. THEOREM:
.6ub.6et.6 06 E wh..(.ch conta..(.n.6 the compact that E
j
..(..6
a
kJR -.6 pace
qua.6.(.-c.omplete. Then
60Jt
en (0) y
polog..(.cal l..(.nea.IL .6ub.6pace
E
1 F
~j .(..6
one has aj(x,v x Sj-l) c ",j(x,.)
E
w.
By
£~(Ej,C~(n)~). that
y, by the definition of bhe to-
sj) - a
A(Cn(Q) y
E
Aj, First we observe
Now let us show the continuity of for any compact
A
I
j
(K
x
, CnCQ» y
sj) is equicontinuous in
and
282
MEISE
c~(n) I, for any convex balanced neigh-
on equicontinuous subsets of
e~ W) ~
bourhood W of zero in
there are
f l' ... , fm
e~ W) such
in
that
Since
f~j): n ~ !~(Ej) is continuous for
there exists a neighbourhood any
k wi th
1
~
sup. yESJ
U of
1 ~ k ~ m, for any
x such that for any
I f~j)
(x)[y I -
x' E U
f~j) (x'}[y I
and any
denotes the gauge of
~j c)
and
y E sj
W. Since
is continuous for any compact subset hence
U
< l.
By our first observation this shows for any
qw
E
k < m
Hence we have for any
where
x'
x E K
x'
Un K
E
S E Y
n.
K of
was
But
n
arbitrary,~jIK is a k
m-space,
is continuous. For any
u
F
E
E
e~(rl)
L
(en(n)
I
eye
,F) the mappingf :=uoi'>: n+F
u
e~P(n,F).
belongs to
It is easy to see that for
u
0
0 < j < n + 1
the mapping
~j
is continuous. Hence we have proved
can
show
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
f (j+1) u
any
{~(L'1j (x + th,y) t
and any 1
n
E
pac.e
Q
o ' It is easy to see that
E be a
nOlL aftY
Let any
paet subset
E
n
F
P (E) C COO
i.e.
F aftd any
.tl.> deftl.>e .tft
(E).
I.>paee
opeft
I.>ubl.>et
n
Ceo W,F) •
Ceo (rI,F), any compact subset K of
E, any
co
f
r!, anyo:::m-
i < n + I, any continuous semi-norm
q on F,
SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
E >
and
0
be given. We shall show that there exists
g
295
E
Pf(E) 0 E
with
P.e,K,Q,q(f - g)
y.6;tem 06 bounded !.>ub!.>e;t
y
a)
Folt any
b)
FOIL
f E C~(!"I,F) ;the oune.;t-Lon f E C~{rI,F) an.d any
any
be.tong!.> ;to
16
2. LEMMA:
.6 ub.6 e;t 06 f(x , l
->-
PROOF:
j
00
f E C~o{rl,F)
wUh f{rl)
be.tong!.> ;to
C G
i2 : E2
continuous linear map
(i~)*(m)[xl
->-
El x E2
.j
=
j
E
IN. Thus
(i~) * : £~o «E I
m([i~(x)l).
: E
~2
by
E CcoWI x
-+
2
.
2
gives rise
~2
x E ) j ,F} ...
to
£~o (E~,F), defined
C~o WI x !"I2'£~o «EI x E2 )j ,F).
Q2,L~0(E2,F}}.Letus
nl the function
f
denote the function
C~oWlx !"I2,F) thatfor
E
gj(x I , , ) :
!"I2'"
L~0(E2,F)
is Gateaux-differentiable and that its Gateaux-derivative is gj+l (~" This proves that for any
xl E!"II
C~oW2,F), hence the function
a
(i J')*0 f(j) is in COO (nIx n ,£s (E J',F)), 2 2 co 2 co'
gj' Then i t follows from
and any xl E
2 .j
by
If now f is any element of C (QIX!"I2,F}, co
Then i t follows from lemma l.a) that
(i~}*of(j)
j
open
(x ) = (0 ,x ) is 2 2 (E x E )j is con2 I i
defined by
then by lemma lob), 1. 5,and lemna 1.c), f(j) is in
any j E lN
;then.
F,
c~o W,G).
g E C (n 'C~o (!"I2 ,F» l
on.e de6-Lne.6 a 6une.;t-Lon
The mapping
(i~)*O f(j)
f(j)
;the 6un.e.;t-Lon
Le;t E , E2 an.d F be l.e.. .6 pac. e.6 and le;t !"I. be an l ~ co E CcoWl x Q ' F) , E. 60lt i =1,2. Then 60lt any f ~ 2
. ),
•
C {rI,L (E,F». y y
tonuous and j-linear, for any
hence
le;t
belong!.> ;to C~W,G)
uo f
j E lN o
obviously linear and continuous, hence
by
E, and
G J....6 a e.lol.led l-Lnealt ;topo.togJ...e.al .6ub!.>pac.e 06
any
xl
00
l.lub.6e;t
be g-Lven.
u E L(F,G)
c}
297
g:
the function
).
f(x ,·} belongs to l
nl ... C~0(!"I2,F), g(x l )
= f(x l ,·)
can
be defined. In order to show continuity of E >
g on
gj
I
let any
xl
E
!"I, any
C"" W 2 ,F) of the fonn Pi K_ Q q co '-'-;,' 2' is uniformly continuous on {xl} x K2 for any j,
0, and any continuous semi-norm on
be given. Since
!"II
298
MEISE
there exists a neighbourhood for any (x ,x 2 ) E {Xl} x K2 l any
j
VI x V2
of zero in
El x E2 such that
and any (h l ,h ) E VI x V 2 2
we have
0.::. j .::. l.
wi th
0
{xl} x K . By 2 f(j+k+l) and (3) it is clear that there exists
satisfying (2). Consequently we have shown that
9 = go
is an
C~O(~l,C~O(~2»'
element of
Linearity and injectivity of
A are obvious. Continuity
of
A
follows immediately from (1) and the definition of the corresponding topologies. Now we want to prove that
A
is surjective if we impose
some
further conditions.
4. LEMMA:
Foft
wb~et 06
E . A~~ume that i
i=1,2
let
let g be a any 6unc.tion -I..n
Ei
be a l.c..
C~O(nl'C~O(n2»'
-t..6 c.ont1.nuou.o. FolC. any
(j,k)
E
and let
~i
bean open
EI x E~ -1...6 a kIR-~pac.e 60ft any (j,k) EJN 2 .
a)
b)
~pac.e
]N2,
any
SPACES OF DIFFERENTIABL.E FUNCTIONS AND THE APPROXIMATION PROPERTY
UYl-t6 oJtm£.y Q
PROOF:
E
Q~
60Jt a.Yly
a) Observe that for any open subset
1. c. space
F
and any
f
E
C~o (n, F)
(x,y)
is continuous on
Kl of
K x Qj
->-
11 of a l.e. space E,any
f(j) (x) [y )
for any compact subset
111 and any compact subset
2
belongs to
Q
l
of
l
x
6~,C(K2
x
Q~))
= C(K ('
This proves the continuity of f J, for any (j,k) b)
E
l
k)
of
11 and
K2
sub-
, the function
in
112 and any com-
Pk«g(j)
(0)
[o])(k»
~) o 1
SuppOhe :tha:t :thelle i.6 a .6ub.6et G 06 :the vec:tOlt .6pace c.on:tbtuou.6 lineall endomOltph-Lhmh 06
E
E' ® E 06 a.U
w-Lth 6ini:te d-Lmen.6iona.e. .i.mage-6,
.6uch that: 1)
The -Ldenti:ty mapp-Lng
belo ng.6 to :the clo.6 ulle 06 G
IE
the compact-open :topology on the vec.toll .6pace a.e..e. c.ontinuou.6 l-Lneall endomoltphihmh 06 2)
Folt evelly
:tha:t
J
E
(f 0 J)
U
and evelly
f E A, it 60llow.6 that :the
Iv =
I V)
f
0
CJ
06
E.
G, evelly nonvoid open .6ub.6e:t V 06
J(V) C U
htlt-Lc.Uon
£CE; E)
6Oil
be.e.ong.6:to the
.6uch Ile-
c..e.Ohufte in
316
A LOOK AT APPROXIMATION THEORY
T
m
06
A
i v.
(Nl)
Fo~
eve~y
x E U,
(N2)
Fo~
eve~y
x E U, Y E U, X
that Fo~
(N3)
f(x) eve~y
~
the~e ~~
~ueh
f E A
~
y,
the~e
~~
f
E
~
0,
the~e ~~
f
E A
o.
~
that f(x)
~ueh
A
f(y) .
x E U, tEE,
t
that
a£
Tt(x)
If
df(x)
(t)
~
o.
E is finite dimensional, conditions 1) and 2) of Theorem 4
are satisfied by
G reduced to
IE. Hence Theorem 4 implies
Theorem
the
Banach-
1.
Condi tion 1) of Theorem 4 implies that Grothendieck approximation property, that is, closure of
E'
~
E
in
E
has
belongs
IE
to
£(E;E) for the compact-open topology.
the Thus
Theorem 4 leads to the following conjecture:
CONJECTURE
5:
FM eve~y g~ven
E, the
6oR.R.ow~ng
m
then
eond-
E. Then
f
E
C (E) belongs
C(E) if,and only if, for every compact sub-
E contained in some equivalence class modulo
0, there is
g
E
W such that
Ig(x)
f (x) I
0
W, for each of which there is It == It (V)
A V, form a basis of neighborhoods at 0; in equiva-
lent terms, when corresponding to every neighborhood
of
U
in
0
W
we may find another neighborhood V of 0 in Wand E > 0 such that co k k Uk=O T (E V) C U. More generally, the members of a collection C of linear operators on
Ware said to be "similarly directed"
neighborhoods
0 in
such that at
O.
V of
T{V)
C It
if
W, for each of which there is It = It (V ,T) > 0
V for every
TEe, form a basis of neighborhoods
Directedness of a linear operator implies its continuity. Both
directedness and similar directedness reduce to continuity when a normed space. These concepts arise only in treating
more
topological vector spaces. Thus the hypothesis in Theorem that the operators in isfied when
THEOREM 6: 6unction~
undelL
the
W is
general
6
below
A be similarly directed is automatically sat-
W is a normed space.
The
pai~
A, W ha~ ~ome ~ep~e~entat~on by cont~nuou~ ~eat
i6 and onty i6
W
i~
A, and the opelLatolLl.> il1
a
A
Hau.6do~66 aILe.
.6pace wh.i.ch
~
toea.Uy convex
.6im.LtalLty dilLected.
A LOOK AT APPROXIMATION THEORY
76 :the paL'!.
THEOREM 7:
Jteat 6unc.tionJ.> and
A,
undeJt
A, W haJ.> J.>ome JtepJteJ.>en:ta:tion by
S iJ.> a vec.toJt J.>ubJ.>pac.e 06
:then :the quo:tien:t paiJt
76 the paiJt
!teat 6unc.:tionJ.>,
76
A,
W
whic.h
:tain
:then
invaJtian:t
Jtep!teJ.>en:ta:tioI1
S i& c.to&ed in
A, W haJ.> &orne !tepJte& en:ta:tion
by
W.
c.ontinuou&
:then J.>pec.:t!tat J.>yn:theJ.>iJ.> hotd& in :the 60ttowing J.>en&e.
S iJ.> a c.to&ed pJtope!t vec.to!t J.>ub&pac.e 06
deJt
c.on:tinuouJ.>
W whic.h iJ.>
haJ.> J.>ome
A/S, W/S
by c.on:tinuouJ.> !teat 6unc.:tion& i6 and onty i6
THEOREM 8:
319
W whic.h i&
invaJtiantun-
S i& :the bt:te!tJ.>ec.:tioI1 06 att c.to&ed vec.to!t /.)ubJ.>pac.('h 06
a!te invaJtian:t undeJt
A, have c.odimen&ion one in
Wand c.on-.
S.
The passing to a quotient statement of Theorem 7 implies
spec-
tral synthesis in Theorem 8, which may be viewed as an abstract version of the Weierstrass-Stone theorem for modules. Let us also point
A is reduced to the scalar operators
W,
then
Theorem 8 becomes the following statement. Every closed proper
vec-
out that, when
tor subspace
S of a locally convex space
all closed vector subspaces of and contain
of
W is the intersection
of
W which have codimension one in
S. As it is classical, such a statement
is
W
equivalent
to the Hahn-Banach theorem. Thus Theorem 8 may be looked upon
as
a
generalization of both the Weierstrass-Stone theorem for modules and the Hahn-Banach theorem for locally convex spaces. We may then ask the following natural question. To what extent the condition of the operators in
A being similarly directed is cru-
cial for the validity of Theorem 6, or Theorem 7, or Theorem 8? Local convexity under
A is not superfluous.
In fact,
reduced to the scalars operators of
W, then it may
every closed proper vector subspace
S of
sll closed vector subspaces of and contain
letting be
A
false
W is the intersection
W which have condimension one in
S, in case W is not assumed to be locally convex.
be
that of
W The
320
NACHBIN
answer to the above natural question is no. The example that I found in 1957 led me to the classical Bernstein approximation problem, asI shall describe next.
EXAMPLE 9: tions on
Let W be the Frechet space of all continuous real funcJR
A = P (JR)
that are rapidly decreasing at infinity. Call
the algebra of all real polynomials on
JR. Every
a E C(lR)
that
is
slowly increasing at infinity gives rise to the continuous linear opera tor Thus
Ta : fEW
->-
which is directed i f and only a is bounded.
a fEW
A may be viewed' as a commutative algebra
operators of
of continuous linear
W containing the identity operator of
W, but each such
operator is directed if and only if the corresponding polynomial constant. It is clear that some
W is locally convex under
w E W vanishing nowhere
in lR such that
Aw
A.
There
of
JR
w E W
that is not a fundamental weight in the sense
B A P - 2 or B A P - 1 below). Then the closure
Aw
in W is a closed
proper vector subspace of W which is invariant under never vanishes in
is
is not dense in
W (this is easily seen to be equivalent to existence of some vanishing nowhere in
is
JR., it can be shown that Aw
any closed vector subspace of
A.
Since
w
is not contained
in
W which is invariant under
A, having
condimension one in W. Thus Theorem 8 does not hold in this case doo to lack of directedness. A fortiori Theorem 7 and Theorem
6
do not
BeJt~.te.i.n
a.pp!tox.-L-
hold in this case for the same reason. This counterexample leads us to the c..e.a..6.6-Lc.a..e. ma.~on
p4oblem, usually formulated in the following two forms, where
P (lRn )
is the algebra of all real polynomials on mn for n = 1,2, ..•. B AP - 1. Let
and
v: lRn
lR+ be an upper semi continuous "weight" n be the vector space of all fEe (lR ) such that vf n 0 at infinity, seminormed by II f II = sup{v(x). I f (x) I ;x EJR }. ->-
CVoo (lRn )
tends to
Assume that
v
v is rapidly decreasing at infinity, that is p(JRn) CCvoo(£).
A LOOK AT APPROXIMATION THEORY
n P(m )
When is
321
n Cv",(m )? We then say that
dense in
v
6uVlda-
is a
me.Vltal we.-ight. We shall denote by S"l n the set of all such fundamental weights in the sense of Bernstein. For technical reasons we also introduce the set Clearly
r n of all such
rn C S"ln
0
such that
v
k
E S"ln
k > O.
for all
This inclusion is proper. n Coo(m )
B AP - 2. Let ing to
v
be the Banach space of all
at infinity, normed by
the special case of
n Cv",(m )
it
is
n WE C(m )
is
IIfll= sup{jf(x)l; x E mn}i
when
v=1. Assume that P(m n ) w
rapidly decreasing at infinity, that is w a we...i.ght. When is
f E C(:nf)tend-
n
P(m ) w dense in
C (mn ),
C
and call
'"
C",(mn )? We then say
that
w
is a 6uVldame.Vltal we...i.ght. If
wE C(IR n )
is rapidly decreasing at infinity, then
fundamental weight in the sense of
~n
vanishes on
and
Iwl
B A P - 2 if and only if
is a fundamental weight in the
B A P - 1. However a fundamental weight vanish on that
n
lR
B AP - I
v
w is a w
never
sense
of
in the sense of B A P - I
rray
and may fail to be continuous.
It
is
in
is a better way of looking at the concept
men tal weights in the sense of Bernstein than
this sense of
funda-
B A P - 2.
The following are the simplest criteria for an upper semicontinuous function
v: m
->-
m+
to belong to
by
r I ' thus to
increasing degree of generality:
BOUNDED CASE: ANALYTIC CASE:
v
ha~
Th e.~e.
a bounde.d a~e.
C > 0
~uppo~t.
a.Vld
c
> 0 60lL wh..i.c.h, 6o~ anlj x E ~,
we. have
v (x)
QUASI-ANALYTIC CASE:
We. ha.ve.
< C • e -cl x'i •
~oo
m=l
I
+
00
whe.Jte,
322
m
NACHBIN
we.6 e.t
0, I , ... ,
In
BAP-I,
P(mn )
the subalgebra
Cvoo(m n ), and we have the weight
C(mn )
of
v in the definition of
Thus
weAflhted
I
n CVoo(IR ). In
P(mn )
B AP - 2, the subrnodule p(mn)w over the subalgebra is contained in
is contained in
of
CORn)
C00 (mn ), and we have the weight w in the definition
was led
app!Lox.[mat.[olll
to
the following general
formulation
of the
pll.obtem. The viewpoint thus adopted embraces the
Weierstrass - Stone theorem for modules, thus for algebras, Bernstein approximation problem. Actually, it is guided by
and the
the idea
of extending the classical Bernstein approximation problem in the same style that the Weierstrass - Stone theorem generalizes
the classical
Weierstrass theorem (see [34] for details). Let V be a set of upper semi continuous positive real functions on
a
completely regular topological space
d.[ll.ected in the sense that, if VI' v
such that
vI'::' A v and
v 2 < A v.
2
E.
v E V
and any
£
Each element of
CVoo(E).
V
is called
f E C(E) such that,
Each
... IIfliv = sup {vex) • If(x) Ii x
f
is
a for
> 0, the closed subset {xEE; v(x)'if(x)1 >d
is compact, will be denoted by seminorm
V
E V, there are A > 0 and v E V
we.[ght. The vector subspace of C(E) of all any
We assume that
natural topology on the we.[gh.ted llpace
E
v E V E}
C Voo (E)
determines a
on is defined
by
the
family of all such seminorms. Let
A
C
C (El be a subalgebra containing the unit, and W C CVoo(El
be a vector subspace. Assume that W is a module over
that
is
appll.ox~ma.t~on
pll.obtem consists of asking for a
description of the closure of W in
CVoo(E) under such circumstances.
AW C W.
The we.[gh.ted
A,
We say that
W is £.oca£..Lzab£.e undell. A .[n CVoo(E) when the following
A LOOK AT APPROXIMATION THEORY
condition holds true: if of
W in
CVoo (E)
f E CVoo(E), then
f
belongs to the closure
if (and always only if), for any
and any equivalence class v(x) • Iw(x) -
323
f(x) I
0
such that
x E X. The I>:tltie:t weigh:ted appltoxi-
for any
ma:tion pILoblem consists of asking for necessary and sufficient conditions in order that We denote by
W be localizable under G (A) a subset of
A in
CVoo(E).
A which topologically generates
A as an algebra with unit, that is, such that the subalgebra generated by
G(A) and one is dense in
We also introduce a subset W as a module of by
G (W)
A,
is dense in
G(W) of
A for the topology of
of
A
C(E).
W which topologically generates
that is, the submodule over W for the topology of
A of
W
generated
C V00 (E) .
A basic result is then the following one.
Al>l>ume :tha:t, 60IL evelty
THEOREM 10:
w E G(W), :theILe il>
y E fl
v E V,
evelty
a E G(A) and eveILY
I>uch :tha:t
v(x). jw(x)i < y[a(x)]
n01t
any
x
E. Then
E
W il> locaV.zable undeIL
A -tv!
CVoo(E).
We may combine Theorem 1'0 with the indicated criteria for membership of
f l ' Let us consider explici tly the analytic case.
COROLLARY 11:
evelty
AI>.6ume :tha:t,
wE G(W), :thelte alte
nOlL
eveILY
C
0
>
and
v E V,
c
>
eveILy
x
E E.
Then
W
i.6 localiza.ble
undelL
and
.6ueh :tha:t
0
vex) • Iw(x) I < C • e-c'la(x)
nOll CUlY
a E G(A)
j
A
-ll'l
CVoo(E).
As a particular case of the above results for modules,
we have
324
NACHBIN
the following ones for algebras. For simplicity sake, assume that is strictly positive, that is, for every that
v(xl
o.
>
caLizabte .in f
E
Let
A be contained in
x E E
there is v E V suen
CV '" (El . We say that A is to-
CV",(El when the following condition
CV",(El, then
always only if)
holds
true:
if
eV",(E) if
(and
is constant on every equivalence class nodulo
E!A.
f f
V
belongs to the closure of
We denote by
G(A) a subset of
A in
A which topologically generates
A as an algebra with unit, that is such that the subalgebra of A generated by
G(A) and one is dense in
A for the topology of
CV",(E).
The particular case is then the following one.
A~~u.me
THEOREH 12: i~
y E r
~u.ch
l
that, 60Jt eveJtIj
x
E. Then
E
E
G(A), theJte
that
v(x)
60Jt any
and eveJtIj a
v E V
A
i~
~
y[a(x»)
toeat.izabte in
eV",(E).
We may combine Theorem 12 with the indicated criteria for membership of
rl. Let us consider explictly the analytic case.
A~~u.me
COROLLARY 13: aJte
and
C > 0
that, 60Jt eveJty v
c > 0
v(x)
60Jt anlj
x
E
E.
We quote
Then
A
~u.eh
[34], (37)
and eve.Jty a
that
< C • e- c • ia(x)
i~
E V
I
tocat.izabte.in
ev", (E)
•
for additional details.
E G(A), theJte
A LOOK AT APPROXIMATION THEORY
325
REFERENCES
[1]
R. M. ARON, Approximation of differentiable functions on a Banach space, in In6inite
Vimen~ional Holomo4phy and Applica tion. 135).
In the complex case, X is regular iff the hull-kernel to:c;>ology is Hausdorff and the :c;>roof relies heavily on the com:c;>actness of the Gelfand to:c;>ology. By
c~oosing
nonarchimedean algebras in which
is not com:c;>act, one obtains counterexamples to "if the topology is Hausdorff, then
each maximal ideal that
U
c W. If
X and let
Since II x (M) II.::. II x II
M}.
M
hull - kernel
X is regular".
U be the unit ball in
Let
M in
U = W, we call
X
W = {x I II x(M) II .::. 1 for every
for
M, it is clear
a V*-algebJta.As will be seen shortly, ( see
the V*-algebras are the nonarchimedean analogs of B*-algebras (2.10». I t is easy to verify ([10], p. 148)
that V*-algebras must be
semisimple.
2.9.
16
.-L~
T
p!ete then
T
a .-L~
O-d.-Lme~~.-Lonal
c-ompac-t
Hau~do~66 ~pac-e
homeomOJLph.-Lc- to the .6pac-e
C(T,F) ul1.deJt the map
M
0
6
and F
max~mal
.-L~
.-Ldea!~
c-om06
~
M = {x E C(T,F) I x{t) t the Gel6and topology. Al~o) C(T,F) ~~ a v*-algebJta ([10], :c;>. 154). In
add.-Lt.iol1., .-Lo
S
t
~~ O-d.-Lmel1.~~o~a.e,
meomoJtph' n)
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
If
f
347
is only continuous, then the convergence of
P tf
to
f
is uniform on every compact subset. It is also well known (1) that there exist Banach spaces such that the bounded and in the space of uniformly ceding result gives a
Cl
several
separable
functions are not
dense
continuous and bounded functions. Thepre-
uniform approximation by H-infinitely differen-
tiable function. For plurisubharmonic functions this kind of approximation gives more or less the same result as proposition 2. Now we shall state the following proposition:
PROPOSITION 3: E and let
v
Le:t
U be a p.6eudo-convex open .6ub.6e:t 06 a Banach .6pace
be a pluft-
I
be .6uch
Then
p •
dA
IAI=p
1.17 COROLLARY:
and
p > 0
be .6uch ~hat
1
nT
Let f E JtS(U/F),
(Cauchy integral formula):
6n f(S)
(x)
~
+
AX E U,
6o~ eve~ A E IC ,
J
= _1_
2 1Ii
f
(~
+ AX) An + 1
I A
~EU, X
I.::.
E E
p.Then
d>'
IAI=p
60ft
n=O,l, ...
1.18 COROLLARY: ~ E
U
and
(Cauchy inequalities):
p > 0
.l p
60ft
~
+ pB C U. Then
~
be .6uch that
Let f E j{'s(U/F), SEcs(F), B E
sup {S(f(x»; X -
n
~
E pB}
n=O,l, ...
1.19 DEFINITION:
A mapping
f: U
holomoftph~c
if for every
¢ E F'
dual of
the function
¢
F)
0
f
+
F
(where
is said to be F'
wea~ly
denotes the
is silva-holomorphic.
S~lva-
topological
THE APPROXIMATION PROPERTY FOR SPACES OF HOI.OMORPH IC MAPPINGS
Let: F be ct .opctee w1.t:h t:he pllopellt:y thctt: 1.6
1. 20 PROPOSITION:
ct eompctct: .oub.oet 06
K 1..0
F, then the elo.oed ctb.oolutelif convex
r (K), 1..0 ct eompctct: .oub.oet 06
K,
357
F.
Then
S1.1va-holomollph1.e mctpp1.ng 1.6 and only 1.6
f : U -+ F f
1..0 S1.1va-holomOllph1.e.
The proof of this proposition follows from Proposition 1.lSand Nachbin [8 I .
1. 21 DEFINITION: A subset
if there is pact in
B E BE If
EB
K
E is said to be a .ot:Il1.et: eompctet set
of
such that
K
is contained in
E is normed, or Frechet (or
£ F l,
strict compact if and only if it is compact in We will denote b y , s of
EB
and
16
(Xs(U;F), 'sl
PROOF:
E.
the locally convex topology on
('0
F 1..0 a eomplet:e loeallif eonvex
BE BE'
U.
.opaee,
t:he.n
is complete, for
S E cs (F) •
be a Cauchy net in (Jes (U;F) ,'s) and
(falunEB)aEI
is a Cauchy net in (X(U') EB;F)"o)
is the compact - open topology). We know that
(X(U') EBi F ) "0)
F complete. Using this fact, it is easy to see that
there is
f E XS(U;F) such that (fa)aEI
(X (UiF),
's).
s
XS(U;F)
1..0 eomplete.
Let (fa) a E I
Then if
com-
K c: E is
then
uniform convergence on the strict compact subsets of
1.22 PROPOSITION:
is
converges
to
f
We now define the notion of Silva-holomorphic mapping of
on
com-
pact type, which will be needed in the next section.
1.23 DEFINITION:
For
linear mappings from of E,
and
E E
-+ I(J (
X
b E F, xl • b E F
I(J
E E'*, where
E to we
denotes the space
of
C, which are bounded on bounded
denote
the
S - bounded More
by
l(Ji E E'*, i=l, •.• ,n, n E IN
E'*
and
bE p', we denote
linear
subsets mapping
generally, the
all
for
S - bounded
358
PAOUES
n-linear mapping
by
The vecto::- subspace of
£b (nE;F)
generated
by all elements of
the
bE F, is denoted
by
form iplx ... xipn ·b, ipi E E*, i =l, ... ,n, and £bf(nE;F). We define the vector subspace be the closure of
£bf(nE;F) in
£b(nE;F), to
£b(nE;F). The topology on £bc(nE;F)
will always be the induced topology by complete space then
£bc(nE;F) of
n
£bc( E;F)
is
a
£b(nE;F). Hence, if complete
space.
We
F
is a define
£bfs(nE;F) =£bf(nE;F) n £bs(~;F) and £bcs(~;F) = £bc(~;F) n £bs(nE;F). For n = 0
we define all these spaces as
1.24 PROPOSITION:
1.25 DEFINITION:
n
£bfS(nE;F)
£bcs( E;F).
is said to be a S.Ltva-boundedn-Une.aJt
A E £b(nE;F)
mapp~ng 06 compact type if and only if Analogously, for
F.
A E £bc(nE;F) .
ip E E*, b E F, we denote the
n-homogeneous polynomial given by
X E
E
+
ip (x) n • b
Silva - bounded E
F
by
.pn • b E P ( n E;F). The vector subspace of Pb(nE;F) generated by all b elements of the form ipn • b, ip E E*, b E F is denoted by Pbf(~;F) • We define the vector subspace
Pbf ( n EiF) in
of
Pbc ( n EiF)
P
bc
a.
n ( EiF) of
Pb (nE iF) to be the closure
n
n
will
P ( EiF). Hence, if b
F
always
is a complet space
is a complete space.
1.26 PROPOSITION: ~nduc.e.6
bc
P ( EiF). The topology on b
be the induced topology by then
P
topolog~c.a.l
The natu~a.l mapp~ng and
•
n
T E £bS(nEiF ) +TEPb(EiF)
algebJt.a.~c. ~.6omoJt.ph~.6m
between
£bcs (nEiF ) and
(nEiF) •
1. 27 DEFINITION:
is said to be a
~va-bounded
n-homogeneouo
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
QompaQ~ ~ype if and only if
polynomial 06
1.28 DEFINITION:
Let
369
P E Pbc(nE;F).
Xsc(U;F) be the vector subspace of
of all Silva-holomorphic mappings f : U ->- F, such that for each 1 ~n n and n E lN, nT IS fix) E P ( E;F). An element f E Xsc(U;F) bc be called a Silva.- holomoJtph-LQ ma.pping a (\
QompaQ~ ~ype
06 U
A main tool of this paper is the notion of £-product by Schwartz [14]
which we want to review.
1.29 DEFINITION:
Given two locally convex Hausdorff spaces
F'c
F, we denote by
the dual of
.c £
(F
I
c'
E)
will
-Ln~o
F.
introduced
and
E
F endowed with the topology of uni-
form convergence on all balanced convex compact subsets of E £F =
x EU
F, and by
the space of all linear continuous maps from
FI
to
C
E, endowed with the topology of uniform convergence of all equicon tinuous subsets of seminorms
S £
£(F~,E),
U
S E cs(F) and
1. 30 DEFINITION:
the
a.pPJtox-Lma.~-Lon
E F',
0. E
lui
PJtopeJt~y,
EB
K of
E,
for all
x E K.
< £,
PJtopeJt~y
there is
and given
v EE', Ivl < o.},
if for every
for all
0.
E
E £F '" F (E.
is said to have
E cs(E), every
K of
E, there is T
£ E
> 0,
E' ® E,
x E K.
A locally convex Hausdorff space E is said to have
the S-a.pPJtox-Lma.tion set
s,
A locally convex Hausdorff space
o.(T(x) - x)
1.31 DEFINITION:
0, there is
such that
K
C
EB
and is compact in
T E E* 0 E, such that
Pa(T(X)- xl < £,
360
PAUUES
1.32 REMARK:
If
S.a.p., E' = E*, and all compact subsets
E has the
of
E are strict, then
E
is a normed space, or Frechet, of
E
has the approximation property. If (En) ~=O
sequence
property, then
E has the approximation property. Hence,
of Banach spaces
E has the S.a.p . .
£F, which has the S.a.p., then E is an inductive limit
Enflo in [3) S.a.p .•
Let E and F be locaU.y convex (ten~oJt
F
E
i~
E-topology) b)
A
pJtoduct 06
E
locaUy convex
and
Hau~doJt66 ~pace
all locally convex
60Jt all Banac.h E
i~
E ® F
Hau~dolt66 ~pace~
matIol'l pltopeJtty i6 al'ld ol'lly i6
16
E
and
F
I.>pace~
6
~pace~.
endowed
F,
~ub~pace
ha~
with 06
E e: F. (E
®e:
§2. THE APPROXIMATION PROPERTY FOR
~ub~et 06
E. Then
locally convex PROOF:
Let
~pace
Xsc(U;C)
601t
ha~
the apPJtou-
F.
E 0 F
E
il.> devt.6e il'l
Hau~doJt66
E £ F,
I.>pace~,
complete
E
®£
F
F denote~ the completion 06 E ®E F) •
Xs(U;(:).
S.a.p. and let
®F
F.
EEF,
We begin our study with an examination of the closure
Let E have the
E
in
and E olt F hal.> the appJtoximation pJtopeJtty, then
2.1 THEOREM:
E
the
F.
aJte locally c.onvex
il.> identical to
Then
the appJtoximation
den~e
be a qua~i-complete ~pace. Then
Let E
d)
Hau~doJt6
a topological vectOlt
pJtopeJtty i6 and only i6
c)
gives an example of
(Schwartz [14 I :
1. 33 PROPOSITION:
E ®
of a
En' which have the approximation
a Banach space which does not have the
a)
if
i~
Ts-den~e
of
U be a balanced
the
open
in
F.
K cUbe a stric compact set. By hypothesis there
is
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
B E
x E K. Let
such that
f E J(S(U;F),£ > 0
that there is
6 >
a.
is the complement of whenever
and is compact in
T E E* ® E
£ > 0, there is
every all
K C U n EB
such that
BE
x E K
and
6 x < distEB(K,CEB{U () E B
< 6.
Since
»,
X
fl
K,
E
U ()
there
{xl' ... ,x } C K. n r > a}).
and
y(x)
=
(B(a,r)
y: K
-+-
sup { 0 x, - PB (x - xi); i
x E K
and
B(x,a) C B(Xi,ox,)'
thus
Now for any
CE (U B
-
E ) B
()
fey) ) < £, continuous
6in.£.tely S-Runge. Then 60~
JCS(UiF)
eve~y
I!>
JC (UiC) ® F
s
locally convex !>pace
363
TS - deMe .in
F.
For the proof of Theorem 2.4 i t will be needed
the
following
proposition, which has important corollaries.
Let: F be a !>pace !>at:.i-66ying t:he 60l10w.ing
2.5 PROPOSITION:
16
t:Ion:
vex. hull
open
K i!> a compact: !>ub!>et:
06
!>ub~et
06
F, t:hen t:he clo!>ed abMfu-tely con-
r(K),.i!> a compact: ~ub!>et:
K,
cond.i-
06
16
F.
U .i~ a vwn-vo.id
06 E, then
n E IN).
PROOF:
Let
for all to
T : JC (UiF) s
f E JC COiF) , s
->-
¢ E F' and
JCS(U;C). for each
be defined by (Tf) (¢) (x) =(¢
JCS(Ui(J:) cF
f E JCS(U;F)
x E U. Clearly, and
by
peg)
p
E
Tf : F~
We now show that the linear map ous. Indeed, let
¢
->-
JC (UiC)
s
fined by
q(¢)
= sup
p( (Tf) (¢»
for all
¢
E
Call it
K C U
g(x) E
sup{1 (¢
F'. Now
0
g
Let
f) (x) I i x
Tf E
A E JCS(UiC) cF (F~)'
L.
q
defined
JCS(UiC)
is a strict
compact f(K)
be the semi norm on
F
is a f
de-
{ I¢(t) l i t E L}. I t follows that
¢ E F'. Hence
Let now fine
F.
belongs
is continu-
set. By hypothesis, the closed absolutely convex hull of compact subset of
f) (x),
F'.
be a TS-continuous semi norm on
= sup {Ig(x) Ii x E K}, where
(Tf) (¢)
0
=F
£(F~i
E
K} < sup{I¢(t) Ii tEL} =q(¢)
JCS(Ui(J:».
= £(F~,
by the formula
JCs(U;C». For each g(x) (¢)
is weakly S-holomorphic, hence
=
(A¢) (x),
x E U, defor
S - holomorphic
all by
364
PAOUES
Proposi tion 1. 20. Clearly, Tg = A, and therefore T is onto JCS(UI(J:) e: F. On the other hand, T
is injective by the Hahn-Banach Theorem.
remains to show that
T
Let II(g)
= sup
is a homeomorphism.
8 E cs(F) and {Ig(x) II
It
K cUbe a strict compact
x E K},
subset.
Let
g E Jfs(UI
we have by the Hahn-Banach Theorem, that
sup {8 ( f (x) ); x E K} = sup { I cp ( f (x) ) :; cp E F I,
sup {
I( ¢
0
I cp I 2. 8,
X E K}
f , Y )I
(8 e: II) (Tf).
This completes the proof.
2.6 COROLLARIES OF THE PROPOSITION 2.5:
6ub-!>et 06 a)
16
U
F
OIL
i-!> a
non~void
open
E, we have:
16
F
,(,6
a complete .6pace and
(Je (UIC),
s
's>
ha.6 the
appILoximation PJtopeJtty, then
In paJttic.ulaJt i6 E ha.6 6inite dimen-!>ion and F i-!> a
com~
plete -!>pac.e, then
b)
16 F ha.6 the appJtoximation pJtopeJtty and c.ondit-ion 06 PJtopo.6i.t.[on 2.5, then in
0)
the
Jfs(U;ati-!>6ie-!>
ha.6 the appJtoximation pJtopeJtty i6 and
JfS(Uipace-!>
F.
i-!>
's-den-!>e in
JfS(U;F),
only
noJt ali Banach
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
The proof of a)
follows from Proposition 2.5 and
1.33 (d). The proof of b) 1.33 (b)i and c)
366
Proposition
follows from Proposition 2.5and
Proposition
follows from Proposition 2.5 and Proposition 1.33(c)
and Proposition 1.22.
PROOF OF THEOREM 2.4: and
Let
K CUbe a strict compact set, S E cs (F)
f E Xs (UiF). By hypothesis, there is
and is compact in satisfying
E
, so that given
B
PB(T(X) - x)
E
for all
< E,
>
B E BE
such that
0, there
where
(X(UoiF), TO)
is
0, there is 'P E I. Since
g E (Esl g E (ES)
I
I
® E*, such that
1:
'Pi®x
i=l Since, for each for of
E. Let
for each
i
,
'PiE (E
s)',
i =1, .•• ,m, 'Pi : ES .... (
'P E E*, where Bi E BE
subset
and for
IIg('P) - 'PIIK
0, there is g E J(S(E;(J:) S E
x E K.
368
PAQUES
It is clear that if
E has the
S. a. p., then
E has the S.H.a.p ..
For the converse it is needed that E be a quasi-complete space, that is, we have the following theorem, which contains the previous theorem for an open subset
which is finitely S-Runge.
be a qua.6i-c.omple.te .6pac.e and le..t
U be an open
2.9 THEOREM:
Le.t
.6ub.6e.t 06
whic.h i.6 6ini.tely S-Runge. Then .the 6ollowing c.onc.f..U;ioYl4
E,
E
U of E,
aJr.e equivalen.t:
ha.6 .the
a)
E
b)
FoJr. eveJr.y loc.ally c.onvex .6pac.e in
S.H.a.p .. @
c)
(JeS(Ui(C), TS) ha.6 .the, appJtoxima.tion pJtopeJt.ty.
d)
E
ha.6 .the,
S.a.p ..
The assumption of
only in
c)
b)
E to be a quasi-complete space is
needed
d) •
+
+
i.6 T s-den.6e
F
JeS(UiF).
REMARK:
PROOF:
F, JeS(UiC!:)
c) is part (c) of Corollary 2.6, which is true for
open subset of
E.
c) ... d) is Theorem 2.7.
remains only to show that
a)
+
d) ... a)
is obvious.
b). This proof is analogous
proof of Theorem 2.1, substituting
g E JeS(Ei(C) @ E
for
any It
to the
T E E* @ E
(cf. Definition 2.8).
2.10 COROLLARY:
S.a.p •
Le.t E be a qua.6.i.-c.omple.te .6pac.e. Then
.i6 and only i6, 6oJr. eac.h
E
ha.6
.the
(Pb(nEi(C), TS) ha.6 .the.
ap-
S.a.p., it follows by Theorem 2.9, that
for
n
E
lN,
pJtox.i.ma.tion pJtopeJt.ty.
PROOF:
If
E has the
any open subset
U of E,
which is finitely S-Runge,
has the approximation property. Since for each
(Jes (U i(C),
T S)
n E lN, (Pb(nEiC),Ts)
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPH IC MAPPINGS
369
(Pb(~;«:)' TS)
is a complemented subspace of (;ICS (U;«:), T S), we have that has the approximation property.
Conversely, in particular, E * having the approximation property, E
has the S.a.p.
2.11 REMARK:
(as in the proof of Theorem 2.7).
By the previous Corollary, we have that
quasi-complete space and S-Runge, then if, for each
(;ICS(U;~),
n
E
IN,
U is an open subset of
E,
if
E
is
a
which is finitely
TS) has the approximation property, if and only ( Pb (nE ; C), TS) has the approximation property.
REFERENCES
[11
R. ARON, Tensor products of holomorphic functions, 35,
[21
Inda~Math.
(1973), 192 - 202.
R. ARON and M. SCHOTTENLOHER, Compact holomorphic mappings Banach spaces and the Approximation property, J. tional Analysis 21,
[31
on
Func-
(1976), 7 - 30.
P. ENFLO, A counterexample to the approximation property
in
Banach space, Acta Math. 130 (1973), 309 - 317. [41
A. GROTHENDIECK, P4oduit4 ten404iet4 topotogique4 et
e4pace4
nucieai4e4, Memoirs Amer. Math. Soc., 16 (1955). [51
c.
P. GUPTA, Malgrange theorem for nuclearly entire
functions
of bounded type on Banach space. Doctoral Dissertation, University of Rochester, 1966. Reproduced by Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brasil, Notas de Matematica, N9 37 (1968). [61
M. C. MATOS, Holomorphically borno1ogical spaces and
infinite
dimensional versions of Hartogs theorem, J. London Math. Soc.
(2) 17 (1978), to appear.
370
PAQUES
[7]
L. NACHBIN, Recent developments in infinite dimensional holomorphy, Bull. Amer. Math. Soc. 79 (1973), 625 - 640.
[8]
L. NACHBIN, A glimpse at infinite dimensional holomorphy,
In:
PJtocce.di.ng.6 on 1no.i.nLte. V.i.men.6.i.ona.t Ho.tomOJtphy, UY!.i.VeM.i.:ty 06 Kentucky 1973, (Edited by T. L. Hayden and T. J. Suffridge). Lecture Notes in Mathematics 364, SpringerVerlag Berlin-Heidelberg - New York 1974, pp. 69 - 79. [9]
L. NACHBIN, Topo.togy on Spae.e.6 06 HolomOlLph.i.e. Mappi.ng.6,Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 47, Springer -Verlag New York Inc. 1969.
[10 J
Ph. NOVERRAZ, P.6 e.udo - e.o nve.xLte., e.o nve.xLte. po .tynom.i.a..te et doma..i.ne.6 d'holomoJtphie en d.i.men.6ion in6.i.ni.e, Notas de Matematica 48, North-Holland, Amsterdam, 1973.
[11)
O. T. W. PAQUES, PJtoduto.6 ten.6oJt.i.a.i..6 de. 6un~oe.6 S.i..tva-ho.tomoJt6a.6 e. a pM PJt.i.edade de apJto x.i.ma~a.o, Doctoral Dissertation, Universidade Estadual de Campinas, Campinas, Brasil, 1977.
(12)
D. PISANELLI, Sur la LF-analitycite. In: Ana.tY.6e oone.t.i.one.t.te. et app.t.i.e.ation.6 (L. Nachbin, editor). Hermann, Paris, 1975, pp. 215 - 224.
(13)
J. B. PROLLA, AppJtox..i.mat.i.on 06 VectoJt Va.tued Function.6, Notas de Matematica 61, North-Holland, Amsterdam, 1977.
(14)
L. SCHWARTZ, Theorie des distributions a valeurs I, Ann. Inst. Fourier 7 (1957), 1 -141.
[15)
M. SCHOTTENLOHER, £-product and continuation of analytic mappings, In: Ana.ty.6e. Fonctionelle et App.ti.cat.i.on.6, (L. Nachbin, editor) Hermann, Paris, 1975, pp. 261 - 270.
[16]
J. S. SILVA, Conce.i.to.6 de 6un~a.o d.i.66eJtene..i.avel em e.6pa~o.6 .tode ca.tmente conveXO.6, Centro de Estudos Matemati:cos Lisboa, 1957.
vectorielles
Appro~mation
Theory and Functiona~ Analysis J.B. Prolla (ed.) ~North-Ho~land PubZishing Company. 1979
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
JOAO B. PROLLA Departamento de Matematica Universidade Estadua1
de Campinas
Campinas, SP, Brazil
1. INTRODUCTION Throughout this paper
X is a Hausdorff space such that Cb(X;l
~
AV(X)
with
the
0 such that vi(x)
L is then equipped
topology defined by the directed set of seminorms
f
..
II f IIv
and it is denoted by
sup {v(x)[ f(x»); x E X} ,
LV",
Since only the subspace we may assume that
L(x)
= Fx
L(x) = {f (x) ; f E L} C Fx is relevant, for each
The cartesian product of the spaces C(X;lK)-module, where
C(XilK)
x E X. Fx has the structure of a
denotes the ring of
all
continuous
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
lK-valued functions on ¢ E C (X; lK)
X,
if we define the product
and each cross-section
(¢ f) (x)
for all
x E
x.
If
WC L
373
f
¢f
for
each
by
¢(x) f(x)
is a vector subspace and
B C C(X;lK) is a
subalgebra, we say that W is a B-module, if BW={¢f;¢EB,f E w}cW. We recall that a locally convex space
E has the applLox.imat.ion
plLopelLty if the identity map e on E can be approximated, on every totally bounded set in
E,
by continuous linear maps of fi-
nite rank. This is equivalent to say that the space
E.
E'
$
E is dense in
ICE) with the topology of uniform convergence on
bounded sets of on
uniformly
E.
Let
p.
If, for each
rna tion property, then
P E cs (E)
I
let
Ep denote the space· E semi-
p E cs (E), the space
Ep has the approxi-
E has the approximation property.
Suppo-6e that, 601L ea.ch
THEOREM 1:
totally
cs (E) be the set of all continuous seminorms
For each semi norm
normed by
£c(E),
x E x, the -6pac.e
the topology de6.ined by the 6am.ily 06 -6em.inolLn-6 the apPlLox.imat.ion plLopelLty. Let
Fx equ.ippedw.ith
{v(x); v E v}
ha-6
B C Cb(X;lK) be a -6el6-adjo.int
and
-6epalLat.ing -6ubalgeblLa. Then any Nachb.in -6pace
wh.ich
LVoo
B-module ha-6 the applLox.imat.ion plLopelLty. The idea of the proof is to represent the space W = LV"" being
I
as a Nachbin space of cross-sect·ions over
X,
£ (W),
where
each
fiber
£(W;F x )' and then apply the solution of the Bernstein-Nachbin
approximation problem in the separating and self-adjoint bounded case. Before proving theorem 1 let us state some corollaries.
COROLLARY 1: Fx
Let X be a Hau-6dolL66 -6pace., and 601L each
be. a nOlLmed -6pace w.ith the applLox.imat.ion plLopelLty.
Cb(X;lK)
be
a -6el6-adjo.int and -6epalLat.ing -6ubalgeblLa.
x E X
Let
let B C
374
PROLLA
Let: L be a vect:oJt .6pace (Xi (F
x) x
CJtO.6.6 -.6 ect:ion.6
peltt:ainirlg
to
.6uch t:hat:
E X)
(1)
06
60Jt eveJty
f E L, the map
x .... IIf(x)1I i.6 upPeJt .6emicontinuoit6
and null at in6initYj i~
a B-modulej
(2)
L
(3)
L(x) = Fx
Then
noJt each
x E X.
L equipped with noJtm
f
1/
1/
= sup {I/ f (x) II i
X
ha.6 the
E x}
appJtoximation pJtopeJtty.
PROOF:
Consider the weight v on
for each 1/
f
1/
x E X.
= sup { 1/ f (x)
REMARK:
Then
LVoo
is
X defined by just
L
vex)
equipped
= norm with
the
Fx ' norm
II i x EX}.
From Corollary 1 it follows that all "continuous sums",
the sense of Godement [6] or [7],
in
of Banach spaces wi th the approxi-
mation property have the approximation property, if the X
of
is compact and if such a "continuous sum" is a
"base space"
C (Xi lK) - module. b
In particular, all "continuous sums" of Hilbert spaces and of C·-algebras, in the sense of Dixmier and Douady [3] tion property, if
have the approxima-
X is compact. Indeed, a "continuous sum"
sense of [3]
is a
COROLLARY 2:
Let X be a Hau.6dM66 .6pace .6uch that
in the
C(XilK)-module.
Cb(XilK) i.6 .6epa.-
Jtat:ing; let V be a diJtect:ed .6et 06 Jteal-valued, non-negative, uppeJt .6emicontinuou..6 6u.nction.6 on
Xi
and let E be a locally convex .6pace
wit:h the appJtoxima.tion pJtopeJtty. Then
CVoo(XiE) ha.6 the appJtoximation
pJto peltty. PROOF:
By definition, CVoo(XIE)
finity, for all
=
{f E C(X;E)
vanishes
at
in-
v E V}, equipped with the topology defined
by
the
i
v f
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
375
family of seminorms
sup {v(x) p(f(x»; x
II f II v ,p where
v E V Let
and
Lv denote
Lv(X)
=
0
by the seminorms
CVoo(XiE) equipped with the topology defined by
or
v E V
Lv(X) = E
is kept fixed. Then, for each x E X, equipped with the topology defined
{V(X)Pi p E cs(E)}. Hence in both cases, Lv(x) has
the approximation property. It remains to notice Cb(Xi~)-modules.
spaces are
X}
p E cs(E).
the above semi norms when either
E
property. Since
v E V
Therefore
Lv
has
was arbitrary, CV",(X;E)
that
all
the has
Nachbin
approximation the approxima-
tion property.
Let X and E be a.6 in COlLollaJI.y 2. Then
COROLLARY 3: (a)
C(XiE)
with the compact-open topology ha.6 .the applLoxima-
tion plLopelLty. Co(XiE) with the uni60lLm topology ha.6
(b)
the
applLoximation
plLO pelLty .
REMARK:
C(Xi~)
In (a) above, it is sufficient to assume that
is
separating.
COROLLARY 4:
(Fontenot [4 I)
.6pace, and let E
plLopeltty. Then
Let X be a locally compact
be a locally convex .6pace wi.th
Cb(XiE) with the .6tltict topology
the
applLoximation
B ha.6 the appltoxi-
ma.tion pltopelLty.
PROOF:
Apply Corollary 2, with
V
{v E C (X i lR);
o
Hau.6d0lL66
v > o}.
-
376
PROLLA
have the
PROOF:
app~aximatian p~ape~ty.
In Corollary 2, take
E
]I 0
be given.
T E leW) consider the map
For each
for
A C W be a totally bounded set.
and let
EX:W'" Fx
is the evaluation map, Le., Ex(f) =f(x),
fEW.
Just notice that
v (x) [ f (x)
For each and for each
x
1 -< /I
f
II v ,
v E V
consider the weight
sup {v(x)[ (U(x»
v E V.
v
on
T ==
(EX
0
T) x E
X defined by
(fl]; f E A}
U(x) E £(W;Fxl. Then
V(X)[ExOT]
for any
for any
T E £ (W), consider the cross-section
v(x)[U(xl]
for every
EX E £(W;F )' since
T E £(W).
sup {v(x)[ (T f)(x)]; f E A}
xi
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
x ~ V(X)[T(X)]
The map
STEP 2:
a~ .i.~6.i.~.i.t!J o~
X,
PROOF:
E
Let
Xo
6aJt each
377
i~ uppe~ ,~emico~~i~uou~ a~d
va~hu
T E £ (W) •
X and assume
Vex )[T(X ») < h. o 0
Choose
hOI
and
h'
such that
(1)
Let
6 = 2 (h" -
there exist
f
l
,f , ... ,f E A rn 2
i E {1,2, ... ,rn}
(2)
6 > O. Since
such that, given
T (A) is totally bounded, f E A,
there
is
such that
liT f - T fillv < 6 /4.
Since
x ->- v (x) [ (T f i) (x)
V , V "",V l 2 rn
for all
X. Let
U
I
= l,2, ••• ,rn).
= V l n V2
x E U
is upper sernicontinuous, there are such that
neighborhoods of
x E Vi (i
Let in
h'). Then
il
•..
and let
n Vrn . Then
U
f E A. Choose
is a neighborhood of
i E {1,2, .•• ,rn} such that
(2) is true. Then
vex) [ (T f) (x)] ::. vex) [ (T f) (x) - (T f ) (x) I + vex) [ (T f i ) (x) I i < liT f < 6/2
h"
T fillv + v(xo )[ (T f i ) (xo )] + 6/4
+ v(xo ) [ (T
f
i
) (x ) 1 o
- h' + v (xo) [ (T f i) (x o )
Xo
1•
PROllA
378
On the other hand, by (1), we have
v (x ) [ (T f.)
o
Hence
~
J -
-
¢2m (u) ,
and writing
(6)
(c)
->- C(u),
(y)
->-
T(u),
we find the normal equations (4) to be equivalent to the relation
T (u) ¢2m (u) ,
whence
(7)
This establishes
->-
C(u)
T(u) ¢2m(u) + E(2 sin ~)2m 2
SCHOENBE RG
404
THEOREM 3:
1~ te~mh
06 the
expa~hio~
1
(8)
¢2m (u) +
whe~e
Qoe66iQie~th
the
6iQie~th
(c.) ]
06 the
E
(2 sin ~ ) 2m
06 the
( ) WvEe
ivu I
WV(E)
holutio~
S(x)
(9)
v=-oo
mi~imum p~oblem,
L: c. M2 (x j ] m
j)
a~e
(10)
l:
j=_oo
Qa~di~al hmoothi~g
We call the solution (9) the
3.
A 6ew
p~ope~tieh
A. in (2.8)
06 the
hpli~e.
s(x) =S(X;E).
Qa~di~al hmoothi~g hpli~e
We have assumed above that
E
> O. However I
if we set
=0
it becomes
I
1
(1)
and a comparison with the expansion (3.28) of Part I, = W
v
B.
the
Qa~di~al hpli~e
S(X) 06
What i.6 the e66eQt 06 the
o~igi~al
sequence
v: Thih .6 ho Wh that
for all
i~te~polati~g
o~
E
(S(v))
data I
(Yv)?
S(x;O) Theo~em
= S(X)
shows ~eduQe.6
that
to the
2.
hmoothi~g .6pli~e
S(x)
= S(x;
E)
This we answer by detennining the "sm:x:>thed"
to compare it with (y ). By (2.9) and (2.10)we find v
(S (v))
ON CARDINAL SPLINE SMOOTHING
406
and therefore, by (2.7),
(S(v»
(2)
+
T(U)
C(u) ¢2m(u) 1
+E
(2 sin .J!) 2m 2
In terms of the expansion
1
( 3)
1 +
E
e
(2 sin .J!) 2m _---.,,....--,-..2:;-_ ¢2m (u)
a~i~e~
(2) shows that the sequence (S(V;E» ~moothing
n~om
ivu
the data (y ) by the v
60~mufa
(4)
S(V;E)
Observe that by (2.8) and (3) the coefficients by
a v (£l
= l: M2m (v j
j)
0V(E)
are expressed
W j (E) •
Is (4) a smoothing formula according to our definition of Part I, Section l? That it is one we see if we inspect its characteristic function
(5)
1
K(u;£l
1 +
E
(2 sin .J!) 2m _ _.,...-......,.....::;2,....-_ ¢2m(u)
for it is evident that
o
(6)
C.
The
< K(u;£l < K(O;£l
~moothing
powe~
1
06 the
for
60~mufa
o
< u < 211 •
(4)
inc~ea~e~
with
E. In [4, Definition 2, p. 53] we gave good reasons
infor
SCHOENBERG
406
the following definition: Of two different smoothing formulae having the characteristic functions
¢(u) and
¢(u), we say that the second
has greater smoothing power, provided that
i¢(u)i < i¢(u)i
(7)
However, if
for all
u,excludingequalityforall
u.
0 < £ < £, it is clear by (5) that
o
< K(U;E)
< K(u;£)
o
if
< u < 27f ,
and the criterion (7l is satisfied. D.
The degJtee 06 exactl1eM 06 the -6moothing 60Jtmula (4) .i.-6=2m-1.
This follows from (1. 7) of Part I, because (5l shows that we have the expansion in powers of
(8)
K(u;
E.
u
£l
1 - £u
2m
+ ...
If we drop our assumption (1.1), and assume only that (Yv'
-L-6 06 poweJt gJtowth, thel1 OlVL COI1-6.tJz.uction 06 the -6moothil1g -6pUl1e 8 (xl = 8 (x; E)
by the 60Jtmutae (2.8), (2.10), and (2.9), Jtema-Ln-6 appl-Lcabte.Of course, its earlier connection with the funtional holds. In fact we will find that sumably, it is still true that our
J(8) =
J(8), of (1.8), no longer 00
for all splines
8(x;£l minimizes
8. Pre-
J(8l, provided
that (Yv) satisfies the condition
of Theorem 2. However, this I was not able to establish. In any case I recommend the cardinal smoothing spline (8 (x; Ell, which represents the modification, found more than 30 years later,of may war-time approach to the problem of cardinal smoothing.
ON CAROINALSPLINE SMOOTHING
407
REFERENCES
[lJ
T. N. E. GREVILLE, On stability of linear smoothing
formulas,
SIAM J. Num. Analysis, 3(1966), pp. 157-170. [2J
T. N. E. GREVILLE, On a problem of E. L. De Forest in iterated smoothing, SIAM J. Math. AnaL, 5(1974), pp. 376 -398.
[3]
FRITZ JOHN, On integration of parabolic equations by
difference
methods, Corrun. on Pure and Appl. Math., 5 (1952) ,pp.155 - 211. [ 4]
I. J. SCHOENBERG, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. of Appl. Math., 4 (1946), Part A, pp. 45 - 99, Part B, pp. 112 -141.
[5]
I. J. SCHOENBERG, Some analytical aspects of the problems
of
smoothing, Courant Anniversary vol ume "SWcUe-6 and E6.6ay.6 ", New York, 1948, pp. 351 - 370. [6]
I. J. SCHOENBERG, On smoothing operations and their generating functions, Bull. Amer. Math. Soc., 59(1953), pp. 199-230.
[7]
I. J. SCHOENBERG, Spline functions and the problem of graduation, Proc. Nat. Acad. Sci. 52 (1964), pp. 947 - 950.
[8]
I. J. SCHOENBERG, Cardinal interpolation and spline
functions
II. Interpolation of data of power growth, J. Approx. Theory, 6(1972), pp. 404 - 420. [ 9]
I. J. SCHOENBERG,
Ca~dinaf
.6pfine
inte~pofation,
Reg.
Conf.
Monogr. NQ 12, 125 pages, SIAM, Philadelphia, 1973. [10]
E. T. WHITTAKER and G. ROBINSON, The eafeufu.6 Blackie and Son, London, 1924.
Department of Mathematics United States Military Academy West Point, New York 10996
06
ob.6e~vation.6,
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla (ed.) ©North-HoUand Publishing Company, 1979
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
M. VALDIVIA Facul tad Pas eo
de Ciencias al Har, 13
Valencia
(Spain)
In [1 I , A. Grothendieck asks if each quasi-barrelled (OF) -space is bornological. We gave an answer to this question in [5 I structing a class of quasi-barrelled (DF)-spaces which bornological nor barrelled. In this paper, in
by
are
con-
neither
the context
of
Kothe's echelon spaces which are Montel, we characterize the
the
spaces
of Schwartz using certain non-bornological barrelled spaces.
As
a
barrelled
consequence, we prove the exis tence of non - bornologi cal (DF)-spaces.
K
of
denote
by
The vector spaces we use here are defined on the field the realor complex numbers. If \l
(E ,F)
the Mackey topology on
(E,F) E. If
is a dual pair, we
E is a topological vector space,
E' is its topological dual. In the sequel and
Ax
A will be an echelon space
its a-dual. Let us suppose that the steps defining
a(n)
(ai n ), a~n), ... ,a~n), ... ), n=1,2, ...
are all positive, they form an increasing sequence index
p,
there exists and index
q such that
each be
n-th whose value for this
is the space generated by 409
for
p
is one. Generally, we follow the terminology of [2 I ~
and,
a(q) '" O. Let
the sequence such that all its terms vanish except
of spaces. In particular,
A
the
kind vectors
410
VALDIVIA
En'
"x[
Here
n = 1, 2,
\.l (" x,
we always consider
a subspace of
,,) I .
Let
P
{In: n = 1,2, ..• } be a partition of the set
tural numbers, such that
In of
F n In
N such that, if
E J, then
j
F E F
is finite, n=1,2, . . . . Let
the set of all the filters on for some
N of
In is infinite, n = 1,2, . • . . Let
filter of all the subsets of tary in
as
I{)
N finer than
F be the
the complemen{F. : j
E
)
F so that, if
M n In f. 121, n =1,2, ..•
na-
be
J}
M
E
F. )
It follows immedi -
ately that, with the relation of inclusion, this set is inductive ordered. Using Zorn's lemma, let
PROPOSITION 1:
Fo.lr. eac.h
U be a maximal element.
n E N, :the .lr.e-6:t.lr.-Lc.:t-Lon 06
U :to
In
-L-6 an
ul :t.lr.a 6LU e.lr. •
PROOF:
Let Al and A2 be two non-empty subsets of
intersects all the elements of
belongs to
Al
U l
and
that
U [U
U and then
{Ip : pEN, P ". n} I
This completes the proof. n = AI· U c N, we denote by "x(U) the sectional subspace of
" x (U) ={a.=(al ,a 2 , ..• ,an' ... ): a. E " x , an =0, } \in E U . U belong to U it follows that U n U belongs to U l 2 2
"x[ \.l ( " x, ,,) I defined by If
such
A n I
U and
For each
n
Al n A2 = 121. Therefore, one of these sets, say Al
Al u A2 = In' and
A
I
and
and, therefore,
L
U
E U}
A CHARACTERIZATION OF ECHELON KOTHE·SCHWARTZ SPACES
is a subspace of gyof
>. x
containing
L is the one induced by
16
PROPOSITION 2:
Let us suppose that the topolo-
]J(>'x,>.).
A i.6 a MOVlte.-i'. .6pac.e. aVld
ab.6oftb.6 the. bOUVlde.d .6ub.6e.t.6 06
PROOF:
I{).
411
I{)
T i.6 a baftfte.-i'. in
~ >,x(N ~ I
n
),
60ft e.ac.h
L, Lt
n E N.
Let us suppose that there exists in
normal subset
B which is not absorbed by
struct a sequence (y ) in q
T. We now inductively ron-
B in the following way:
that we have already obtained the elements
Let
us
suppose
in
Yl'Y2, •.. ,yq
B such
that
~
a
r EN(p)
where
N(l), N(2) , ... ,N(q) N (1)
joints, such that
are finite subsets of
=
In' mutually dis-
In which does not lie in l-:l(p -1), being
N(l) U N(2) U ... U N(r). The space
I{)
n >.x(N ~ I
n
) is the topological direct sum of
I{)
Let
Bl be the projection of
B2 be the projection of
B onto
B onto
normal set it follows that
E2
El
n >,x(N ~ (I
according to
according to
E
l
B
l
. Since
B is not absorbed by
can find an element
E
~ M(q))).
n
2
, and
. Since
Bl U B2 C B. Moreover, Bl + B2
is a bounded subset of the finite-dimensional space sorbs
,
contains the first elerrent of I , and N (I ), p> 1, n p
contains the first element of M(r)
E K, P = 1,2, .•• , q
r
yq+l E B2 C B
T,
neither
such that
y q+ 1 ¢ (q + 1) T.
E B
2
l
B is ~
B.
let a Bl
, hence T ab. Therefore, we
412
VALDIVIA
can be written in the form
The element
l:
a
rEN(q+l)
where
£
r
N (q + 1) is a finite subset of
r
In' disjoint from
each
set
N(l), N(2), ... ,N(q) and that it contains the first element of In which is not contained in partition of
I
n
M(q). The sets of the sequence (N(q)) define
. Let
U{N(2q-l):q
U {N (2q)
Since the restriction of an
U E U
such that
U on
U () I
n
q
In
1,2, ... }
1,2, ... } .
is an ul trafil ter, there
coincides with
say. Therefore, Y2Q E "x(U) , q = 1, 2, ...
PI or
The space
relled, because is a sectional subspace of sorbs the set
"xl
exists
P 2 ' U n I n =P l , "x(U)
jJ (" x,,,)
is bar-
li hence T ab-
{Y ,Y , ... ,Y ... } and it contradicts 2q 2 4
Y2q ¢
( 2q) T, q = 1 , 2 , . .. .
is
Since the normal hull of every bounded subset of bounded, i t follows that
PROPOSITION 3:
16 "
1.6
T absorbs everyboundedsubsetof k + 1
Since
o
lim i-+ co
Let
11 be the set
[2, p. 421] .
{m ,m , ... }. Obviously, M 2
1
finite set. Let us suppose that we have constructed I
so that
I
If
I
P numbers
n
p
I
n M
p
¢,
r
p
11
is an
subsets
inof
is an infinite set and
~
r,
p, r
l,2, ... ,q .
{r ,r , ... ,r , ... }, suppose also that there are two natural l 2 i k + p, i so that k > p p a(k) r.
ark) r.
1.
lim
a(k+p) r.
i-+ oo
cp
~
0,
lim i .... co
1.
1.
0,
i > i
(k ) a p r.
P
1.
Let
H
q
= U {I
p
:p =1,2, ... ,q}. If we arrange the terms of
H nM as q
a sequence
we obtain, for
u >
that
p=1,2, .•. ,q,
lim
0,
i-+ oo
From (1) and the condition of space, it follows that
i
> i
P
(1) •
A[~(A'AX)] not being a Schwartz
416
VALDIVIA
M~H
q
is an infinite set and the sequence
does not converges to zero. Therefore, we can select (t ) of (si) and a positive integer i
kq+l > k + q + 1
(k) at. cq +l
(k+q+l) at.
i-+oo
"I 0,
l.
lim
(kq+l) at i
i-rco
l.
Let element of ti tion
subsequence so that
(k) at.
l.
lim
a
o•
be the set {t , t , ... , ti ' ... } together with the first 2 l N which does not lie in
P = {In: n = 1,2, ... } of
H . In this way we obtain a parq
N such that
In is infinite,
whose
properties will be used in the sequel.
THEOREM 1: i~ il1
>.. x[ fl
16 (A x,
the Mantel
~paQe
>..)] a den~ e ~ ub-6 pace
G
which i-6 baltlt elled a.nd non bOlt-
l101og,[ca.l.
PROOF:
Using the number
construct
the space
and the subspace
k and the parti tion
L as we did at the beginning of
and the vector
a (k). We will prove that
nological. Let
T be a barrel in
[3, p. 324], hence
..)] which is the linear hull of
G of
bounded subset of
P obtained above,
0 on
:R, and q > 0 on [O,l]}.
In fact:
16
n < m + 1, then.
1.
PROPOSI,",ION:
2.
PROPOSITION: Foft a.Le.
a.ll
fEC[O,l]
m, a.n.d
~+n .
a.dm,.[tl.> bel.> t a.ppftO x,.[ma.t,.[o n.1.> to
424
WULBERT
III. CHARACTERIZATION AND UNICITY OF APPROXIMATIONS FROM
R~, the idea
As in the characterization of approximation from
is to change the problem to that of approximation from a more computable set. We will first state a special case so that the general case
a
will appear less absurd. Suppose that
1)
m
~
E
no common factors and that the degrees of
a
,
that a and
and
b
b
have
are such
that
2n + aa < ab + m. Let H(a,b) = {h
(3.1)
where
3.
M
ab + m
~~
a
and
~
PROPOSITION:
ze~o
PM: sgn h(x) = - sgn a(x)
E
~~ a be~~ app~ox~ma~~on ~o
Now in general suppose
b .::: 0).
a
and
~
a
f
- 1)
E
~
•
6~om
E
Z (b)}
f
~6 and onty
b have no common quadratic factors a
f.
~6
H (a,b) .
From the definition of
However it may be possible that
real zeros. Let
x
Z(f) denotes the zero set of a function
be~~ app~ox~ma~~on ~o
we may assume that
for
and b
have some
F be the greatest monic common divisor of
a
~ (i.e. common and b.
Put (3.2)
a IF
and
b
o
b IF .
Now put (3.3)
For
M
ba
E
Q.m n
max { abo + m, aa
+ 2n}
we now define
Z (b ) n JR (3.4)
o
Z(a,b)
{
if
2n + aa < ab + m
IZ':olOlRlU'.lUl-.lif
2n+'a"b+m
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
425
Por convience we will write f(oo)
(3.5)
for
lim f(x) x+oo
f (_00) for
and
lim f (x) , x~-oo
when these limits exists. Now define: H(a,b)
(3.6)
{Ph: h E PM : sgn h (x)
for
4. COMMENT:
x
E
- sgn a
o
(x)
Z(a ,b)}.
With the above notation proposition 3 above
is
still
valid. Our interest in proposition 3 is that one can compute the number of possible sign changes of members of H (a,b)
and
use
this
to
derive an extremal alternation type of characterization for approxi-
~.
mations from
However the result separates into many cases de-
pending on the number and parity of the pOints in and in
Z(a,b) () [1, 00).
Rather than presenting
Z (a,b)
the
() (- 00 ,0]
complicated
statement of the alternation theorem, we will give some of the consequences.
5.
COROLLARY:
6.
COROLLARY:
and Z(b) () m
Be6t
=
~6
app~ox~mat~on6
6~om
a~e
un~que.
Supp06e a, and b have no c.ommon 6ac.toM, m + db > 2n + Cla a ¢ . Then 1) b ~6 a be-6t app~ox~mat~on to f E C [
°,
a.nd only
in
f
-
a
b
2 + max {m + ab, 2n + da}.
7.
COROLLARY:
A c.on-6tant
6unc.t~on ~6
a be6t
app~oximat~on,
to
a
426
WULBERT
Qontinuou~
an
8.
6~om
6unQtion,
ext~emat atte~nation
16
COROLLARY:
r E ~
IV.
n
06 tength
a Qontinuou-6 6unQtion f
and
(i)
r i-6 a
(ii)
-r
be~t app~oximation
i-6 not a
APPROXIMATION FROH
06
f
be~t app~oximation
but to
f - 2r.
Rm( m
0
g E C~(lRn)
gp,
I
)
belong
mo E :IN such that
there exists
for any
2
. From this follONS
p E P(lR n )
contains
the sum of any two such products and also any polynomial.
The Be.fLMte-tn .6pac.e on
DEFINITION 1: B
when
lRn,
n = 1 I is the complex vector space
a be a seminorm on
closure of
of
all
Bn or simply
products
gp,
p E P(lRn ).
g E c~(lRnj, Let
denoted by
Bn
A
I
C
Bn. Then
A in the seminormed space (Bn/a)
DEFINITION 2:
A semi norm
a
ii. a
will denote the
:= Bn,a
Bn is po.tynom-ta..t.ty c.ompa..t-tb.te if the
on
module operations (g/f) E COO (lR n ) o
x B
fEB
n are continuous. SPC(lR )
n,a
when
Let
a
.... fEB
n,a
will denote the (directed) set of all poly-
nomially compatible seminorms on
EXAMPLE 1:
.... gf E B n,C!.
n,C!.
Bn.
be an -tnc.Jt.ea6-tng .6em-tnoJt.m on
I f I 2. I g I • Then
a E SPC (lR) .
Since
Igfl 2. IIg III f I then
I f I
I fl·
a(gf) 2.
Bn
that is a(f) 2.a(g) 00 n In fact, let g E Co(JR ), f E Bn.
IIg lIa(f). Also
ali) = a(f) sinoe
It is clear that finite positive linear combinations of increasing seminorms are also increasing seminorms. EXAMPLE 2:
Let
m E :IN * ,
ak ,
i k I
-
13 (f) :5. ( f E C~(mn).
13IB n
for all
k,
described
is of the type
is as in the proof of oo ,
n
E be the vector space
Ukd f vanishes at infinity for all
em' m =1,2, ..•
13(f - emf)
for all
Ikl :5. m, be a family of weights on m.
k
such that
Ikl~m
=
in Example 2. I f 1, then
IN*,
E
Proposition
fEE. Also
max II ~ II) II film Ik,:5.m
Thus the seminormed space (E,13)
satisfies
the
hypothesis of Proposition 2.
EXAMPLE 6:
Let
m and
~,
notes Lebesgue measure on Given
p,
1 < P < + "", let
tributions 13(f)
=
f
on
R
n
ik
I :5. m, be as in Example 5. If
m.n , let
d]Jk = Uk d A
such that
scribed in Example 2. Furthermore
I k I < m.
for all
akf
E
.cP(]Jk)
13i Bn Cm(mn) c
for all k, Ik
is also
of
is dense in
the E.
of this fact is similar to that used in proving density Also
de-
E be the vector space of all complex dis-
Ikl;m (JlakfIPd]Jk)l/p. Then
spaces [11].
d A
in
i
:5. m,
type deThe proof Sobolev
(3 (f)
Once again the hypothesis of Proposition 2 are satisfied.
Let
PROPOSITION 3: 1)
16
i3
E
a.
be a 6uI'ldamel'ltal ~em'£l'loJtm on
n SPC(m )
6undamenta.l.
.{.4 4uc.h that
Rn.
i3 < constant -0., then
i3 '£.6
FUNOAMENTALSEMINORMS
Let
2)
lR \ { 0 }, Xo
E
IR n and
E
S(f) = a(fo I, satisfies conditions
(N)
i~ a ~el6-adjoint
~ub
if 1), 2) and 3) above are true.
LEMMA 2:
Le.t
a
E
SPC(lR n ).
16
A C COO (]Rn)
o
algeblta that hati...o6i..eh condi..ti..onh (N), then A i..h den.oe in
PROOF:
From Proposition 1 it is enough to show that
B n,a
436
ZAPATA
Further, since the topology defined by T,
we need only show that the closure of
A in the
is weaker than
latter
topology
n
c~ (IR ) •
contains If
C~
C is a subset of
cmn) we will denote by C its closure
in the topology T. Assume also that if
n c~ (IR )
on
Cl
gl, ...
,ge E C
are real and
C is a subalgebra. In this case,
hE cooCill-)
is such that
h(O)
-
then
G = {gl (x), ••• ,g-e (x), x E mn} is bounded in
the Weierstrass approximation theorem for differentiable we can approximate stant term, since
h on h(O)
G
= O.
i = 1, .•. , t. Hence
k E mn,
in the topology Let
by polynomials
pEP (IRt)
Furthermore, akg,
functions,
wi thout
is bounded
1-
lRi,using
p (g l' ... , 9 t ) approximates
f E Coo(lRn )
c
f 'I 0
be given. Assume
and
Al
h (g l' ... , 9 t)
let
H
is a subalgebra of
satisfies conditions (N). In particular, for any
Also
be its
x E H
Since
A and also
there exists
hE C"'(lR) such thath~O, h(g(x» >0
g(x) '10. Choose
such that
h(O) = O. From the above remark, it follows that hog
is posi ti ve on a neighbourhood of
ness, there exist Let
all
T.
is a self-adjoint algebra, then
g E Al
con-
for
support. Let Al be the set of real parts of functions in A.
and
n
h(gl, ... ,g-e) E C. In fact, it is clear that h(gl, ... ,g-e) E CoOR).
Since the set
A
= 0, 00
hlEC""(lR)
bourhood of
O.
gl E Al
and
be such that If
r >0
hl=l
x.
(r,+co),
,
K.
Hence
on a neigh-
Hand
fl E Al
has compact support, say
K. Then from Nachbin's Lemma, there eXl.st gl, ... ,gi f = h(gl, ... ,gi) on
on H, gl(O) =0.
hl=O
on
1
fl
since this is a closed subalgebra. Also, fl
such that
gl~r
such that on
hog E A . l Hence by compact-
E~,
eo
i
hE C (lR), h(O} =0
f = fl 'h(gl, ... ,gt)
mn and from the remark on subalgebras it follows that
f
E
on
A, since
h(gl, ••• ,g-e) E A. Now the proof is complete.
DEFINITION 5: defined by
For
z E
Q;\
IR, let
9 z be the complex function on
lR
FUNDAMENTAL SEMINORMS
1 x -
Cl
Let
Cl
'
x
E
lR.
gz E C~ (JR) .
It is clear that
LEMMA 3:
Z
437
r6
E SPC (JR).
gz E P (lR) a
6o!L
z E C \ JR, theYl
,6ome
6uYldameYltal.
~,6
PROOF:
Let
In fact, for
E = P (lR) m=O
true for some
U •
We claim that
g';P (JR)
C
E
for all
mE IN •
this is evident. Assume that the proposition
mElN.
Let
pEP(lR).Since
is
q=gz(p-p(Z»EP(lR),
then from the assumption it follows that
g~+ 1 (p
Now the mapping
Since
-
p ( z) )
f E Ba ~ g~f E Ba
is continuous hence
E is a complex vector space we have
g~+l(p _ p(z» + p(Z)g~+l E E.
So the claim is proved. Further, the mapping
f E Bu ~
h E '~s se If -a d'Jo~n , tsi P (JR) ' con t ~nuous, ence nce for all
•;s. So
is
E BCl
(-)l t EE gz =gz
l E IN ; whence
m, n
for all
From this it follows that the complex algebra gz
f
is contained in
tions (N) since
E.
{gz}
Also
E
IN .
A generated by gz and
A is self-adjoint and satisfies condi-
satisfies conditions (N).
From
Lemma
2
it
438
ZAPATA
follows that
P(lR)
Let
LEMMA 4:
is dense in
a E SPC(IR),
Ba' that is, a is fundamental.
z, z' E
a: \
IR. Then thelte eX-i..6t.6 a pO.6ilive
pEP (lR) .
PROOF:
Let
pEP (JR).
Since
g zp
it follows
From the definition of
a(gf) .::. CllgII
a, there exists
m
cdf),
for all
m
k~O
C > 0
and m E IN such that
g E C~(lR),
fEB.
k! r k+l
To finish, it is enough to observe that the number C ,=1+ iz-z' i CC z,z z does not depend on
THEOREM 1:
Let
p.
a E SPC(IR).
In
thelte e.x..i...6t-l>
z
E
a: \
lR
.6aeh :tha.t
a i.6
6andamen..ta..e
the .6et in eomplex. plane
i.6 unbounded, then a i.6 6anda.mental. Convelt.6ely i6 then
PROOF:
Po. (z) i~ unbounded
Assume that
60ft al!
z E C \ IR.
Pa(z) is unbounded. Let
p E P(lR)
be such that
FUNDAMENTAL SEMINORMS
~
a(gi P )
q
then
1
and
P (m)
E
p(z)
439
O. If
,
gzp
and
= p(z)
q - gz
By choosing a constant Cz,i > 0
.
as in Lemma 4 it follows that
c Z,l..
Since
P a (z)
is unbounded, then
gz
P (IR) a
E
and from Lemma 3
a
is
fundamental. Conversely assume that n
IN*
E
that
be given. Since
a(gz - p)
~ ~
a(gzq) = na(gz - p) ma 4 it follows that
Pn
Then
E
. ~
g
z
Let
a
is fundamental. Let
E
p(m)a,
there exists
q
= n(l
(x - z)p).
1. If
Ci,z
a (gl.. q)
< C.
P (m) , a ( g i Pn)
~
is
a
a:: \ m
E
pEP(JR) Then
q
E
and such
P (JR) and
positive constant as is !em=~. C.
To finish we let
1.,Z
1
z
1.,Z
n
and
Hence
-C-.-
Pa(z)
is
1.,Z
unbounded.
Let
THEOREM 2 (quasi-analytic criterion):
a
1
~
SPC(IR).
E
+
16
00
n=l
PROOF: on
Let
P (lR).
T be a continuous linear form on Let
B a
such that T vanishes such
that
on D. In fact assuming this,
from
D denote the set of complex
numbers
z
Imz < 1. Define h(z)
It is enough to prove that
T(gz)'
h =0
zED.
440
ZAPATA
- - a.
Hahn-Banach theorem it follows that a.
is
zED, n E IN.
If
n > I
~ z
T vanishes on
is also true for
Ih (z) I
n =
P (m)
o.
zED. Then
then n-l _x_ _ zn
-
it follows that
h (z)
Hence
zED,
for all
0
a, there exist
a.(gf) ~clIglim a(f)
+ l) !
and
g E C~(m),
zED
n E IN.
m E lN such that
fEB.
we have that
zED, n E IN.
for all
Let
(z - zo)gzgz
