P-ADIC ANALYSIS AND MATHEMATICAL PHYSICS
Series on Soviet & East European Mathematics - Vol. 1
P-ADIC ANALYSIS AND MATHEMATICAL PHYSICS V. S. Vladimirov, I . V. Volovich and E. I . Zelenov Steklov Mathematical Institute Russia
Vfe wh
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First published 1994 First repint 1998
P-ADIC ANALYSIS AND MATHEMATICAL PHYSICS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfromthe Publisher.
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ISBN 981-02-0880-4
Printed in Singapore.
CONTENTS INTRODUCTION
ix
Chapter 1 ANALYSIS ON T H E F I E L D OF p-ADIC NUMBERS I.
The Field of p-Adic Numbers 1. p-adic norm 2. p-adic numbers 3. Non-Archimedean topology of the field Q of p-adic numbers 4. Quadratic extensions of the field Q 5. Polar coordinates and circles in the field Q ( V ? ) 6. Q and § 7. Space Qp
9 12 13 16
Analytic Functions
17
1. 2. 3. 4. 5.
17 20 22 23 28
p
p
P
p
1
II.
III.
IV.
Power series Analytic functions Algebra of analytic functions Functions e", l n ( l + x), s i n r , cosa: Theorem on inverse function
1 1 3 5
A d d i t i v e a n d M u l t i p l i c a t i v e Characters
30
1. 2. 3.
30 34 37
Additive characters of the field Q Multiplicative characters of the field Q Multiplicative characters of the field Q (\/£) p
p
p
Integration Theory 1. 2. 3. 4.
38
Invariant measure on the field Q Change of variables in integrals Some examples of calculation of integrals Integration in tQ^ p
r
38 39 42 48
vi
Contents
V. T h e Gaussian Integrals L 2. 3. 4. 5. 6.
54
The Gaussian integrals on the circles S The Gaussian integrals on the discs B The Gaussian integrals on Q Further properties of the function A (a). Example Analysis of the function S(a, q)
55 65 67 68 72 77
V I . Generalized Functions ( D i s t r i b u t i o n s )
78
1. 2. 3. 4. 5. 6. 7. 8.
y
y
p
p
Locally constant functions Test functions, n = 1 Generalized functions (distributions), n = 1 Linear operators in V Test and generalized functions (distributions), n > 1 The direct product of generalized functions The "kernel" Theorem Adeles
V I I . C o n v o l u t i o n a n d the Fourier T r a n s f o r m a t i o n 1. 2. 3. 4. 5.
79 81 84 86 89 90 91 92 94
Convolution of generalized functions The Fourier-transform of test functions The Fourier-trans form of generalized functions The space L Multiplication of generalized functions
94 99 106 110 112
V I I I . Homogeneous Generalized Functions 1. Homogeneous generalized functions 2. The Fourier-transform of homogeneous generalized functions and T-function 3. Convolution of homogeneous generalized functions and 6-function
116 116 122
4. Homogeneous generalized functions of several variables
134
2
131
Chapter 2 PSEUDO-DIFFERENTIAL OPERATORS ON T H E FIELD OF p-ADIC NUMBERS IX. T h e O p e r a t o r D ° 1. The operator D ,a £ - 1 2. Operator D~ a
l
143 144 147
Contents
3. 4. 5. 6.
a
Equation D i> = g Spectrum of the operator D in Q a > 0 Orthonormal basis of eigenfunctions of the operator D Expansions on eigen-functions a
P I
a
X. p - A d i c S c h r o d i n g e r O p e r a t o r s 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
r
a
r
p
P
2
a
p
154 156 158 165 167
Bounded from below selfadjoint operators Compactness in Lj{Q") The operator a' + V Operator D", a > 0 i n B Operator D , a > 0 in S Operator D" + V(\x\ ), a > 0 in Q , p / 2 Operator D" + V{\x\ ), a > 0 in L (Q ) (p f 2 ) The lowest eigen-value Aq Operator D" + V(\x\ ), a > 0 in Q , p ^ 2 (continuation) Example. Potential cv > 0 (p ^ 2 ) Operator D + K ( | x | ) , a > 0 outside of a disc (p ^ 2 ) Justification of the method of Sec. 1 0 . 1 1 Further results on the spectrum of the operator D" + p
vii
p
p
a
P
14. Non-stationary p-adic Schrodinger equation
167 170 171 176 179 181 183 184 188 190 192 195 198 200
Chapter 3 p-ADIC Q U A N T U M THEORY XI. p-Adic Q u a n t u m Mechanics 1. 2. 3. 4. 5. 6. 7.
204
Classical mechanics over Q The Weyl representation Free particle Harmonic oscillator Lagrangian formalism Feynman path integral Quantum mechanics with p-adic valued functions P
X I I . Spectral T h e o r y i n p-Adic Q u a n t u m Mechanics 1. 2. 3. 4. 5.
Harmonic analysis Operator theory The theorem about dimensions of invariant subspaces Study of the eigenfunctions Weyl systems and coherent states
205 207 210 212 215 220 224 226 227 229 229 236 241
viii
Content!
6. Symplectic group 7. Investigation of eigen-fu net ions for p s 3 (mod 4) X I I I . W e y l Systems. I n f i n i t e D i m e n s i o n a l Case 1. 2. 3. 4.
Weyl algebras Positive functionals Fock representation Equivalence of L-Fock representations
X I V . p - A d i c Strings 1. 2. 3. 4. 5. 6. 7.
Dual amplitudes p-Adic amplitudes Adelic products String action Moduli space and theta-fu net ions Multiloop amplitudes Rigid analytic geometry and p-adic strings
X V . q-Analysis ( Q u a n t u m G r o u p s ) a n d p - A d i c A n a l y s i s 1. p-Adic and q-integrals 2. Differential operators 3. Spectra of the g-deformed oscillator and the p-adic model
250 253 262 262 263 265 270 272 272 275 278 281 282 284 287 290 290 292 292
X V I . Stochastic Processes over the F i e l d o f p - A d i c N u m b e r s 1. Random maps and Markov processes 2. Brownian motion on the p-adic line 3. Generalized stochastic processes 4. Quantum field theory
293 293 297 299 300
Bibliography
302
References
309
INTRODUCTION Since the Newton and Leibnitz time differential equations over the real number field have been used in mathematical physics. I t is not customary to discuss why exactly real numbers should be used. Why has this happened? The point is that physical processes take place in space and time and spacetime coordinates are usually considered as real numbers. Since the Euclid time the three-dimensional Euclidean space R has been treated as the physical space. As i t is known an important development of this point of view has been done by Riemann and Einstein using the Riemannian geometry, but basically up to now IR is a mathematical model for space and ]R is a model for space-time. 3
3
4
3
These ideas have become so common, that IK is perceived as the true physical space. But in fact the Euclidean space R is not more than a mathematical model for the real physical space. I n order to convince ourselves that the Euclidean space is a good model for the physical space geometrical axioms from elementary geometry should be checked in practice. To this end lengths of segments, angles etc. should be measured precisely. However in quantum gravity and in string theory it was proved that there is the following obstacle to perform such measurements . I f A i is an uncertainty in a length measurement, then the inequality 3
A*>' ' = / p r
(i)
P
takes place. This inequality is stronger than the Heisenberg uncertainty principle. Here h is the Planck constant, c is the velocity of light and G is the gravitational constant. l i is called the Planck length and it is an extremely small quantity, approximately 1 0 cm. We are not going to discuss here a derivation of the fundamental inequality (1) which has a long history. Let us emphasize that by virtue of (1) a measurement of distances smaller than the Planck length is impossible. p
- 3 3
IX
X
p-Adic Analysis and Mathematical
Physics
Let us turn our attention to axioms of Euclidean geometry. I n a list of axioms there exists the so-called Archimedean axiom, which was at first pointed out and analyzed by Veroneze and Hilbert. According to the Archimedean axiom any given large segment on a straight line can be surpassed by successive addition of small segments along the same line. Really, this is a physical axiom which concern the process of measurement. Two different scales are compared in this axiom. I t means that we can measure distances as small as we want. But as we just discussed, the Planck length is the smallest possible distance that can in principle be measured. So a suggestion emerges to abandon the Archimedean axiom at very small distances. This leads to a non-Euclidean and non-Riemanian geometry of space at small distances. How can one construct a physical theory corresponding to a non-Archimedean geometry? As it is well known there is an analytical description of geometry. One uses coordinates to describe a geometrical picture. There are two equivalent approaches geometry
•—•
number system.
The usual Euclidean geometry is described by means of real numbers. I f we want to abandon the standard geometry for description of small distance in physical space-time we have to abandon real numbers. What should be used instead of real numbers? In computations in everyday life, in scientific experiments and on computers we are dealing with integers and fractions, that is with rational numbers and we never have dealings with irrational numbers - infinite nonperiodic decimals. Results of any practical action we can express only in terms of rational numbers which are considered to have been given us by God. Certainly, there exists generally accepted confidence that i f we carry out measurements more and more precisely, then in principle we can get any large number of decimal digits and interpret a result as a real number. However this is an idealization and as it follows from the previous discussion we should be careful with such statements. Thus, fef us take as our starting point the field Q of rational numbers. A geometric notion of distance corresponds to a notion of a norm on Q . Norm is a real valued function |x| with the following properties 1) |z| > 0, |a>| = 0 4=> X = 0, 2) M = |z||t/|,
Introduction
xi
3) l * + y | < | * | + |»| for any rational numbers x, y. What norms do exist on Q? There is a remarkable Ostrowski theorem describing all norms on Q. According to this theorem any nontrivial norm on Q is equivalent to either ordinary absolute value or p-adic norm for some prime number p. p-Adic norm \x\ is defined by the following. Let us fix a prime number p = 2 , 3 , 5 , . . , . Any rational number x can be represented in the form x = p " ^ , where v is an integer and m and n are integers which are not divisible by p. This representation is unique. Then by definition P
\x\ = 1/p- .
(2)
p
At first sight this definition looks artificial but according to the Ostrowski theorem there are no others nonequivalent norms on Q, I t is easy to verify the validity of the properties l ) - 3 ) for \x\ . In fact instead of triangle inequality a stronger inequality takes place 3') \x + y\ <max{\x\ ,\y\ }. p
P
p
p
A norm that satisfies 3') is called non-Archimedean. The completion of the field Q of rational numbers with respect to usual absolute value leads to the field IR of real numbers. Analogously, the completion of the field Q with respect to p-adic norm \x\ leads to the field Q of p-adic numbers for any prime p. p
p
Any real number can be represented as a decimal, that is a series ±10"
£
6„10-
n
•
(3)
0 2
()
2
fl
P
p
for rational numbers k, t and x. Thus the wave-function of free particle, the so-called plane wave can be represented as a product of plane waves of padic particles. It can be interpreted by saying that an ordinary free particle consists of p-adic ones, like an elementary particle consists of quarks. Formulas like (9) are called Euler or adelic products. A well known example of such formulas is the Euler representation for the zeta function
p
*
An analogous representation takes place for propagator in the field theory:
and also in p-adic string theory. Elaboration of the formalism of mathematical physics over p-adic number field is an interesting enterprise apart from possible applications, as i t promotes deeper understanding of the formalism of standard mathematical physics. One can think that there is the following principle. Fundamental physical laws should admit a formulation invariant under a choice of a number field. Thus we include into consideration not only rational, real and p-adic number fields but also other fields. There are a number of other formulations of quantum mechanics and field theory which are equivalent over the real number field, but they are different over Qp.. In Euclidean formulation of p-adic quantum mechanics one uses an action S=
j[ I or \x\ < 1/p) forms the principal ideal of the ring S . Obviously this ideal has the form pZp. The residue field 2 p / p 2 consists of p elements. In multiplicative group of the field Z / p S there exists a unity 77 / 1 (for p ^ 2; for p = 2 n = 1) of order p — 1 such that the elements 0, n , n , . . . ,rf~ = 1 form a complete set of representatives of residue classes of the field Z / p 2 (see [128,121]). p
P
p
p
p
p
2
v
p
p
E x a m p l e s . For p = 3, n = — 1 ; for p = 5, n = 2. As numbers 0 , 1 , . . . , p — 1 form also a complete set of representatives of residue classes of the field S / p 2 p then from representation (2.1) it follows the second canonical form of any p-adic number x ^ 0; p
2
x = p->W(x' + x' p + x' p - ...) 0
1
X. = 0 , 1 , 7 ) , . . . ,rf~\
3. Non-Archimedean
2
,
r
x^O,
0=0,1,...}
Topology of the Field
Q
p
.
of p-Adic
Owing to the inequality 3) of Sec. 1.1 the norm on the field Q the triangle inequality \x + y\
p
< max(|x|p,|t/|p) < \x\ + \y\ , p
p
x,y £ Q . p
Numbers p
satisfies
6
p-Adic Analysis and Mathematical
Physics
Therefore in Q one possible introduces the metric p{x, y) = \x — y\ , and Q becomes a complete metric space. From representation (2.1) it follows that Qp is a separable space. Denote by B (a) the disc of radius p with center at a point a € Q and by S (a) its boundary (circle): p
p
p
1
y
P
y
7
B (a)
= [x:\x-
a\ < p ] ,
S (a)
= [x :\x-
a\ = pT\,
y
y
It is clear that B (a) =
a p
B - {a)cB (a), y 1
y
S (a)
= B ^ a A ^ . j ^ . S ^ a ) C B .{o),y
B {a)
= \ J 5 (a), f |
y
y
y
fl»
v
l' 7 ; then Afi = M f l S y M = M n f Q p ^ ) . 3) | a | = p = | & | . Let 7
p
2
2
t
T
T
V
p
2
7
p
p
7
3
7
a = p " ( a + aiP+O2p + . ••),
7
,
a* ^ o , |a - fc| - p * . Then t
p
(Q \S _ _ {a)).
a
+ ...) 7 -
where On = Jrj, Oi = t i , . . . , a * - i = Mi = M n B - * _ i ( ) , Mi = MC
2
6 - P" (6o + & i P + hp
0
p
T
i
•
1
Lemma 2 asserts that any set of the space Q which consists of more then one points is disconnected (see [174,186], I n other words, a connected component of any point coincides with this point. Thus Q is a totally disconnected space. p
p
By following the proof of Lemma 2, for the case when the set M consists only of two points, we can see that there exist disjoint neighborhoods of these points. I t means, that the space Q is Hausdorff. p
L e m m a 3. A set K C Q and bounded in Q .
p
is compact in Q
p
if and only if it is closed
p
• Necessity of conditions is obvious. We will prove their sufficiency. As Q is a complete metric space then it is sufficient to prove countably compactness of any bounded closed (infinite) set K (see [237]), i.e. that every infinite set M C K contains at least one limit point. Let x G M, then \x\ = p-T( > < C (M is bounded), so 7(3;) is bounded from below. p
E
p
Let us consider two cases. 1) j(x) is not bounded from above on M. Then there exists a sequence {sfc.fc — 00} C M such that 7(211) —• 00, k —* 00. I t means that | i | = p ~ * —* 0, k —* 00, i.e. n —* 0, t —* 00 in Q and 0 £E K. 2) 7(2) is bounded from above on M. Then there exists such number 70 that M contains an infinite set of points of the form 7
p
P
p ( i o + n p + •••).*> < *i < v7 o
Km #
0
As xn takes only p— 1 values then there exists an integer an, 1 < an < p— 1, such that M contains infinite set of points of the form p (ao + X\p+ ), and so on. As a result we obtain a sequence { a j , j ' = 0 , 1 , . . . }, 0 < <JU < ya
8 p-Aiic Analytit and Mathematical Physics 7 o
2
p - 1, a ^ 0. The desired limit point is p ( a + &ip+ a p + ...)€ (K is closed). 0
0
2
K •
Corollaries. 1. Every disc B (a) and circle S (a) art compact. 2. The space Q is locally-compact. 3. Every compact in Q can be covered by a finite number of disjoint discs of a fixed radius (see Corollary 3 from the Lemma 1). 4. In space Q the Heine-Borel Lemma is valid: from every infinite covering of a compact K it is possible to choose a finite covering of K. y
y
p
p
p
E x a m p l e 1. The circle S can be covered by (p — l j p " discs B y ( a ) , 7 > 7', with the centers
7 - 7
y
7
r
)
a = p " ( a + 0.1P + •-. + a _ _ p ' " ' ' 0
7
v
_ 1
1
0 < a, ( >{l + a 2 + 1
2
a !
2 + ...)
J
2 + ... (4.4)
and thus a\ ~ a = 0. 2
Sufficiency. Let a satisfy the conditions 1) and 2). Let us construct a solution of Eq. (4.2). We put 7(3;) = (1/2)7(0.).
Analyait on the Field of p-Adic Numbcri 11 Let p ^ 2. From ( 4 . 3 ) it follows that a number xo has to satisfy the conditions Q = a (mod p), 1< < p— 1 , X
0
1
Such x exists as 1 < a < p— 1, ^^J ^ = 1. From ( 4 . 3 ) it follows also that 0
0
numbers Xj, J = 1 , 2 , . . . have to satisfy the conditions 2xc,Xj = a, + Nj
(mod p),
0 < x, < p - 1
(4.5)
where integers Nj depend only on a J b , % - l . Therefore numbers i j are successively denned (uniquely) from Eq. ( 4 . 5 ) as 2XQ is not divisible by PLet p = 2. From ( 4 . 4 . ) it follows the equation t
a
^ * _
( l + l )
1
I
+
x
2
(
m
o
d
2
)
which is always solvable for (13 = 0 , 1 . From ( 4 . 4 ) it follows also that integers Xj, j = 3 ) 4 , . . . have to satisfy the conditions xj = o , j i + jVj +
(mod 2 ) ,
xj = 0 , 1
(4.6)
where integers jVj depend only on Xi,xs,... ,£j—i- Therefore numbers Xj are successively defined (uniquely) from Eq. ( 4 . 6 ) . • Let n be unity which is not a square of any p-adic number, i.e.
= 1,
This fact we shall write as n g Q * . (For p ^ 2 it is possible to take as 17 2
the unity introduced in Sec. 1.2.) C o r o l l a r i e s . 1 . For p ^ 2 numbers £\ = n, £2 = p, £3 = pn are not squares of any p-adic numbers. 2 . Every p-adic number x can be represented in one of the four following forms: x = Cjy where y <E Q and£o = 1, £ 1 = n, £2 = p, £3 = pn (p / 2 ) . 3. There exists only three non-isomorpkic quadratic extensions of the 2
p
field®? :® ( /ej),j = 1,2,$ (p? 2)4 . For p = 2 every 2-adic number x can be represented in one of the eight following forms: x = e^y where y £ Q2 and £0 = 1 . £1 = 1 + 2 = 3, e = 1 + 4 = 5, £3 = 1 + 2 + 4 = 7, e = 2, £5 = 2 ( 1 + 2 ) = 6, £ = p y
2
2
4
6
2 ( 1 + 4 ) = 1 0 , £7 = 2 ( 1 + 2-1-4) = 1 4 (or equivalently Sj = ± 1 , ± 2 , ± 3 , ± 6 ) .
12
p-Adic Analysis and Mathematical Physics
5. There exists only seven non-isomorphic quadratic extensions of the field Q : Q ( E 7 ) , j = l , 2 , . . . , 7 . 6. The quotient group \Q consists of four elements €j,j = 0, 1,2,3 for p ^ 2 and of eight elements £j ,jj — 0 , 1 , . . . ,7 for p — 2. 2
2
V
2
p
p
Note that for p = 3 (mod 4) as a number n can be taken —1 because (see [204]) =
( y )
(
1
-
,
" = -
1
;
x(\x\ ,\y\ ) p
(4.8)
p
2
is valid. Jn particular, from x + y = 0, it follows that x = y = 0. • I t is sufficient to verify Eq. (4.8) only for the case \x\ = \y\ . I f it would be | i + j/ |p < \x \ = \y\j, then the congruence x% = —yl (mod p) would be solvable and thus —1 would be quadratic residue modulo p which contradicts to the formula (4.7). • p
2
2
p
2
p
5. Polar
Coordinates
and Circles
in the Field
Q (y/e) p
2
Any element of the field 0. Then p
p
0
= y, x 0
:
P
(1 - ^ 7 )
=
> 0 •
= y,,... ,x _i = y ^ ;
x
u
> kip E **p~ * i 0(*)= E *(*-!)---(*-« + 1 ) / ^ * " " ,
(2.1')
n 0, 1 < fc < P - 1- Then 0
m
=
In & — In &o In i > 1— Inp Inp
and thus
=
lim
p
* =
*—co,JteZ+
lim
p
t—eo,*eZ
'!"
= 1.
•
+
By using the formula (1-3) to the series (2.1) and the relation (2.3) we obtain the equalities (
,
(
n ,
' • ( / " ) = >-{/) = r ( / - ) ,
0 = 0,1,... .
(2.4)
From here and from Lemma 3 of Sec. 2.1 it follows that i f p p then R(f) = p and 7 + 1
7
< r(f)
) = R(f) for some «o > 1; 2) R{f)
= P
1 - 1
O I
n > no
7
then either the
(2.5')
then either the equalities (2.5) are valid 1
/£(/) = R(f) for some «o > 1-
= R(fW),
= p
(2.5)
= mjW), P
n > n
0
(2.5")
22
p-Adic Analysis
and Mathematical
Physics
The formulas (2.5) can be interpreted by the following way. I t is possible to differentiate and to integrate an analytic termwise any times; by differentiation a radius of convergence may increase in p times, but by integration this radius may decrease in p times. As we see the situation somewhat differs from the case of real numbers. 3. Algebra
of Analytic
Functions
We denote by A a set of analytic functions in the unit disc B . Such functions are the only ones denned by the series (1.2) for which the condition \fk\ — • 0. k -*• oo is fulfilled (see Sec. 2.1). The set A is linear over the field . On the set A we introduce norm ] | / | | by the formula 0
P
p
11/11 = f « J / * l p .
/ € A
0
(3.1)
The functional (3,1) is in fact a norm, besides the non-Archimedian • Let ll/H = 0, / € A, i.e. max \f \
= 0 then f
t p
one.
= 0, k = 0 , 1 , . . . and
k
hence / = 0. Let a G Qp, a £ 0. Then M
= max|o:/*| - | a | m a x | / , | p
P
p
= | | ||/||. a
p
Finally, i f f,g 6 A then +
= m a x | / +g \ t
t p
< max max
| , p
|„] < max[||/j|, | | | | ] . ?
•
Theorem. The space A is a Banach algebra. • Prove completeness of A. Let a sequence {/",« —* oo}, / " £ A be fundamental. As fn
" -n\=™jft-fjr\p then the sequences {/£, n —. oo) are fundamental for every fc = 0 , 1 , . . . , thus they converge to some f G Q uniformly with respect to k (see Sec. 1.3) and hence k
lim f -co
k
p
= lim l i m f? = l i m lim f
k
t—oo n--oo
n—-oo I—.oo
= 0.
Analysis on ihe Field of p-Adic jVumiera
23
Therefore the function
/*»= J2 h> 0\
,
=
P
n
p
\f'(*)\p\* -a\p=P + -
1
Hence f(x') = y € S + (b) D U{b). Therefore the power series for the inverse function g(y) = g>(b)(y-b) + ... p
n
a
+
1
converges at the point y € S (b) and thus in the disc B +„(b). Reducing i f necessary the radius p we achieve that the function g(y) also will satisfy the equality (5.3) p+n
p
p+n
|s(y)-a|
P
= ^
\y-b\ , p
y€B . p+n
From here it follows that the inclusion 9(B (b)) p+n
C B (a)
= g(U(b))
fi
which together with (5.4) gives U(b) = B + (b)p
n
•
30 p-Adic Analytit and Mathematical Phyiics I I I . A d d i t i v e a n d M u l t i p l i c a t i v e Characters The field Q is an additive group. multiplicative group. P
1 . Additive
Characters
We denote by Q* = Q \ { 0 } its P
of the Field
Q
P
Additive character of the field Q is called a character of additive group Q , i.e. a continuous (complex-valued) function x( ) defined on Q and satisfied the conditions Ixt )! — 1> P
x
P
P
1
x(* + y) = x(*)x(y).
1
*.S>GQ .
1
C - )
P
a
n
a
Analogously one defines (additive) characters of the field Q ( i / £ ) of subgroup By, y G S, of the group Q . It is clear that every additive character of the field Q is a character of any group B . The function vp(^) = exp(2 ri{^} ) (1.2) p
P
P
y
1
p
for every fixed £ G Q is an additive character of the field Q and the group By. I t follows from the relation for fractional parts (see Sec. 1.2) P
P
{x + y}
p
= {x}
p
+ {y} -N.
AT = 0 , 1 .
p
Our goal is to prove that the formula (1.2) gives a general representation of additive characters of the field Q and the group B . Let x( ) arbitrary additive character. From (1.1) we have the relations x
P
x(0) = l ,
x(-x)
= W)
i
= x- (*l
D e
a
n
7
x H - ^ W f ,
n£Z (1.3)
At first we investigate characters of the group B . Let x £ 1 be such a character. Prove that there exists k G 7L such that y
X(a0 = l ,
xGB
t
.
(1,4)
• By virtue of the conditions x(0) = 1, \x(x)\ = 1 and x(i) is a continuous function on B i t is possible to choose such branch of the function y
Analysis on the Field of p-Adic Numbers 31 l n x ( z ) = t a r g x ( i ) that it will be continuous at 0 and argx(O) = 0. In particular there exists k £ S such that | a r g x ( x ) | < 1 for all x 6 B . Taking into account that nx e Bt for all x £ B and n £ S+ we conclude from (1.3) that k
k
|argx(x)| = |-argc(na0| < n n
nEZ ,
;
x G B
+
k
and thus arg\(a:) = 0 and x(x) = l , x € B .
•
k
We assume that the disc B in (1.4) is maximal so that as x(x) ^ 1 in then k < y. Now we prone for any integer r, k < r < 7, the equality k
B
y
p
+i
v ( p - ) = exp(27rimp-' ),
7
3m = 1 , 2
p "* - 1 ,
(1.5)
where m does not depend on r, • For r = y i t follows from (1.4) and (1.3) as 1 - X(P-*)
k
r
= x(p-^- )
p1
= [x(p- )] ~' •
For k < r < y r
X(p" ) = X ( p -
r +
y
y
^ ) = [X(p- )}" "
= [exp(2 rimp-
7+i
7
k
k
+k
T
r
)]" " .
•
7
Denote £ = p m where | £ | = p ~ * | m | > p- p--> = p " and | f | < p~ . Then owing to (1.2) the representation (1.5) takes the form Xp(p~ ) X p ( p 0 > and thus owing to (1.3) we have p
p
p
k
y
=
-1
T
X(P~ )
r
X (p" O,
=
k p . Let now x 6 By\B . The following representation is valid p
p
k
{x)
X
= x {Sx),
• Let x £ B-y\Btx = x p~ 0
r
+ xip"
P
3£GQ , P
Ki >P" P
7
-
(1.7)
Such x can be represented in the form r+1
+
x - pr k+1
h+1
+ x',
x' G B , k
xo^Q
32 p-Adic Analysis and Mathematical Physics for some r, k < r < 7. By using (1.6) and (1.4) we get the representation (1.7): r
i o
r + 1
xO) = [ x { - ) ] [ x ( p P
r
k+1
)r'...
r+1
= [xp(p- or\xp(p- tw° r
P
+ x - t:
ap
k+l
•-•
T+1
= x {z - t
k+i
[x (p- z)r'- xp(*'o P
+ ... + ^
lP
i
\x(p- )r'-+ x(*')
x
k
+
l
P
-
x
k
+
i + x'i) = x {*0 P
=
The case £ = 0 is impossible otherwise x( ) contradicts to the definition of the number it.
1
X (0) = 1 in B p
T
• which •
Hence we have just proved that any additive character of the group B has a form (1.2) where either^ = 0 or | £ | > p ~ ' . Now let x(0 ^ 1 be an additive character of the field Q . Then in a disc Bo it is represented in the form y
T + 1
p
p
( 0
x(*) = x « < ° M .
£ 'eQ ,
P
> 1•
P
(1.8)
We shall prove that in the disc Bi i t is represented in the form ll)
x(«) = Xp(Z *),
=t
w
-Kb,
3*0 = 0 , 1 , . . . , p - 1
(1.9)
• As S i = Bo U Si and Bo (~l Si = <j> v/e shall prove at first the representation (1.9) in the circumference S i . Any point a: £ Si is represented in the form 1
x = p- x
+ x',
0
3 x = 1,2,... , p - l ,
x' *) P
for some £0 = 0 , 1 , . . . ,p — 1. The representation (1.9) is valid also in B owing to (1.8). •
a
Ano/yiij on ike Field of p-Adic Numbers 33 Continuing this process we obtain in the disc B )
x{*)=xpie *),
2
the representation
+&+&i>
for some t]\ = 0 , 1 , . . . ,p — 1, and so on. As a result in Q representation (1-2) X(x) = Y ( £ ; E ) ,
we obtain the
p
£ = ? + & + Sip + - • • € Q .
p
p
Hence, any additive character of the field Q
has a form (1.2) for some
p
x
s
a
In other words, the mapping £ —• Xp(£ ) ' homomorphism of the additive group of the field Q onto the group of additive characters. This mapping is one-to-one (i.e. from the equality Xp{£i ) = Xp(& ) f ° " x e Q it follows that £ = £3). p
x
p
x
r
a
L
Now we have T h e o r e m . The group of additive characters of the field Q is isomorphic to its additive group Q , and the mapping £ —* Xp{£ ) gives this isomorphism. p
x
p
Let us denote Xoo(i) = e x p ( - 2 ^ ) ,
x e R .
(1.10)
Then the following (adelic) formula is valid
II
X P ( « ) = 1.
*eQ.
(l.ii)
2
Analysis
on tke Field
of p-Adic
I t is necessary to consider three cases: z = J),np,p, where n Sec. 1.4). We shall prove mt%,
n t % w
m
vt%,
Numbers
Q*
•
P
37
a
(see
(2.8)
which means that Q* / Q*. it
2
2
Let conversely prj G Q* , i.e. prj = a — nb for some a,6 £ Q , 6 ^ 0 . But the last equality is impossible for none of a and 6 ^ 0 . Indeed rewriting it in the form iP
we see that for 1) \y\
p
p
< 1 the number 1 + j - is a square of a p-adic number
(see Sec. 1.4) and then pn would be a square, 2) | y \ > 1 the number 1 + ^ j p
is a square of a p-adic number and then n would be a square, 3) \y\p = 1 is impossible. Other statements (2.8) are proved similarly. Now we prove that Q " ^ Q"*. Let z = n. Then rj G QJ by the Lemma of Sec. 8.2. Let c = p,pv. Then - e G Q as i t is represented in the form (2.5) with a = 0 and b = 1. • (
>fl
P i £
R e m a r k . For p = 2 (2.6) takes the form rank ( Q J / Q " ^ ) = 2,
rank ( Q ^ / Q ^ ) - 4 ,
see [37] Sec. 6. 3. Multiplicative
Characters
of the Field
Qp(i/£)
According to Sec. 1.5 every element z of the field Q p ( v ^ ) is represented in the form either z ~ r p
|»o(o')| = 1 ,
0
«' € S, 0
«eC.
But C(a) > 0 and therefore w (a') = 1. Hence C(a) = | a | £ ~ \ a e C. We shall find a number a. As 5o is a union of disjoint sets pBo + k = 5_i(fc), k = 0,1,... ,p — 1 whose measures are equal then measure Bn — p measure B-i and hence d(xp) = ^dx i.e. C(p) = * = | p | . Thus a = 2, C(a) — |a| and (2.1) is valid. • 0
p
p
0 p-Adic Analyait and Mathematical Physics By using the formula (2.1) to an integral we get j f(x)dx
= \a\ j
f(ay + b)dy,
p
^ 0 .
•
Let us show how to use this formula to do simple integrals. Example 1.
j dx = p , y
y£Z.
It follows from the formulas (2.1) and (2.2) that j
j
dx=
I«l»'| = \t\ \x'\ > p and therefore Xp(i ') ^ 1. Then performing the change of variable x = y — x' we get that the desired integral is equal to zero: p
p
p
x
j
X
p
m d x =
B-,
j
,
Xp{i{y-x ))dy
j {cly)dy.
= (-ix-) Xp
Xp
B,(* - 7 + 2, y € S
N
p
2
2
(-2exy)dy
Xp
+ p+...+
0
(**_, -
Xl
1
~ y ^ - i j p ^ "
yelp*
- 7
+ •• - ? \ Jp
2 - L(y ,y\,... 0
-
, yN+1-2)
-e xlyx -,-i 0
+
where L does not depend on yN+-,-i- Then taking into account the formula (3.13) of Sec. 4.3 for I = JV + 7 - 1 we have for the integral
p-
N
£ E
E
-
E 2
exp f - — £ i y w 0
+ 7
-p(2-i) -i J= 0 .
Analysis on tke Field of p-Adic Numbers 57 Acting similarly we are convinced that in cases y > 1, | x j ft p* the desired integral equal to 0, and in the case 7 > 1, | x | = p i t is equal to p
1
p
j
2
fcOO
- y) )dy
S ,y =*o T
Xp
S ,yo=x ,... p
X
M*
- y?)dy
= •••
•S,,yo=io,yi=ii
0
j =
J
=
J
2
(e(x
~ y) )dy
dy
,y-,-i=x -,
y
0
7
-( -l)-l
y
7
=
l
i
owing to the formula (3.13) of Sec. 4,3 for I = 7 — 1. E x a m p l e 2 . p # 2, |e| = 1, 7 € Z (see [215,218]) p
2
P^l-^Xpiep* ),
*lp < P "
7 + 1
, 7P >
Ap(erp)Vp\
4>=P >
2
T
0,
3
7 = 1. 7>2,
7
x|p^P , 7 > 2 . (1.2)
• Similar to the Example 1. Some peculiar are: For 7 = 1, | x | < 1, y € Si we have p
2
2
|ep(x - 2 x y ) | = | e | | p | | i - 2 x y | p
P
P
p
2
< ^ max(|*| , |2xj/| ) < i max( 1,p) = 1 . p
Therefore 2
2
2
X (cp(3! - y) ) = Xp(epy )x [£p{z p
P
- 2xy)} = x (epy) . p
But {spy }
= | i ( e + £ip + .. .)(yo + yip + • • • ) | a
p
0
7
i
V P 2
— ) -ex 27ri{epz } P / P
1
For 7 > 2, j « | = p , y £ S p
7
p
2
j
a
0
V
O 2, | i | H*
= 2 \ y e S we have
2
7
= {2
- yfh
_ 2 T
( 1 + M + . . . ) [ ( * , - !fi)2 + £ i - y )4 + .. . ] } 2
2
a
.^37-4) - ^ ( ^ i - ai)»3r-8 1
= L(y ,yi,... 0
and the desired integral is equal to J
xM*
f
- y?)dy =
xM*
2
- y) )dy
= •••
y i=»i, sg =13
•S„yi=ii
/
v-.-J = ' i - a
For the last integral we have a
{ e ( * - y) >2 =
+ Ei2 + . • • H K - j - y - i ) +
- *,)2 + . . . ] '
7
and hence the desired integral is equal to y
=
w»
2 , r i
x
(^+Y)( 7-i-!/T-i)
e x p ^ i Q + ^ ^ T - i - ^ - i ) 2«t k=0,l
a
;
dy
1
+
(M)]—h(l t)]
= l + i ( - i ) * * = v^A (e) . 2
Analysis on ike Field of p-Adic Numbers 61 Here we have used the formular (3.13) of Sec. 4.3 for I = y — 2.
•
E x a m p l e 4. p = 2, |e|a = *> 7 G Z (see [239,218]) M» 2 - » + ,
-1,
Ma < I .
7=1,
1,
Ma = 2,
7=1.
3.
2
(1.4) • Similar to Examples 1-3. Special cases are the following. For 7 = 1, |z|2 = 4, y G Si we have
2
{2 (x - y) ) £
2
= j ^(1 + 2 + £ 4 . . . )[1 + (xi - 1)2 + (x - )4 £ l
3
a
(«, " l )
1
+
4
+
2
Vi
+
, *1 -
2 '
and the integral is equal to
= exp Si
1 + i. .•"(-ir
a
= A (2£) . 2
•4
62
v-Aiic
Antlytii
and Mathematical
For 7 = 2,
N
= 1 ,N 3, |ir| = 2"", 7 € S we have 2
2-
Z l + 3
7
( l + E I 2 + - . - ) [ ( * ! " » ) + (as - » ) 2 + . . .
+ («7-a + !/7-3)2 " + • - • + (Z2-,-4 - y 2 7 - 4 ) 2 " + .. / 7
4
21
5
and the integral is equal to
*T*1"1
r
!.-l- T-l
In the last integral we have {2e(*-y)% = | j ( l + e 2 + c 4 4 " ) [ « T - a - 1/7-2 + 1
2
= £ [('7-3-Vr-a)*+4(«T-i - * r - i ) ' +
- ^-1)2+.. . ] J 2
- * r - i ) ( « r - a - 1*7-2)]
Analyst! on the Field of p-Adic Numbers 63 and the integral is equal to
S ,yi — c,,...
/
JTI, e x p
- irT-3) +
y
E
y
P< j[(z7-2-!/7-2) + («7-l-S/7-l!
- s f y _ i ) ( % _ - y - 2 ) + (2ci + e ) ( : r , _ - y - 2 ) ] !
3
e
!
= 1 + e" + 2 e x
7
A
k
i +^
2
+
e
7
2
2
*)* ]} '
[ — ( 1 + 2ci + e ) = 2 ^ < - l ) ' ' i ' > = 2A (2£)
P
a
a
E x a m p l e 5. p # 2, | o | > p
2 - 2 7
p
, 7 € Z
l/
I
4
a
E E *p { T l * + ^ + *=0,1; = 0,1
_ j X (a)\a\; \ bx)dx = p
+
y-i-i)
3
e X
E
+ 4(%_i
Xp(ax
2
-
,y _)=T -
y
=
3
\
(-£),
p
(1.5)
0,
s. 2N
• Let a = o~p~ where either a = e or a = ep, |e| = 1 (see Sec. 1.4). Under the condition |aj > p ' ' , either N > 1—7 (for c — t) or N > 2—j (for IT = ep). Performing in the integral change of variables of integration p~ x = y, dx = p~ dy we get p
2 2 1
p
N
N
I
2
Xp
(p- "cTX
7
+ bx)dx
S, J
N
= p~
2
b p aw 2
=
N
P~ Xp
N
Xp(vy -rl>P y)dy
4(T
)/
6p
64
p-Adic Analysis
and Mathematical
Physics
Using the formulas (1.1) for a = e, N + 7 > 1 and (1.2) for IT = ep, N + 7 > we get for the desired integral the expressions 2
P
~
N x
> ("in")
I
=
M*P)VP
I 0
^
I S
= P
W
+
7
. « =
*
i f otherwise.
By combining these cases and taking into account that 1 = A (0,
P
P
= K\
N
A (n) = A M , p
N
P \^\;
!
p
p
N
- ^ = \a\ >\ p
,
= \a\ H p "
< p, and we use the formula (1.2) for 7 = 1,
then ^ ~ I r
P
b \ \
3
{
I
6 p~ '
, + s
Y
> p , and we use the formula (1.2) for 7 = 1, 2
then
2
L
(2.1)
27
2
2
• For |a|pP < 1, y € S we have | a x | = | a | k | < 1 hence X ( a x ) = 1, and the formula (2.1) follows from the formula (3.1) of Sec. 4.3 7
P
P
p
2
P
j x (bx)dx By
7
2 7
m
7
= p fi(p |6|p) .
P
2
2 7
2
Let now H p ? > 1, \a\ = p " \ JV = 1, 2 , . . . , a = s p " " , |c|p = 1, Performing the change of variable of integration x = p ~ y, dx = p->~ dy, \y\ = p ~ \x\ < p we get p
N
y
N
N
p
j
N
P pT, i.e. \h\p > \a\ pi = p™~\ we have \p - p. Taking in account that in (2.2) \e\ p ' ' > p , i = 1,2,... ,N we conclude owing to the formulas (3.1) of Sec. 4.3 and (1.7) that all integrals i n (2.2) are equal to 0, and the formula (2.1) is proved in this case. p
p
y
N
N
2 1
p
2
P
For | £ I P < P , i-e- I P * (2.2) takes the form (2.1): 7
7
"
^
1, \a\ = p ~ ~ \ N = 1,2,..., a = e p ^ " ^ , = 1 is considered analogously, owing to the formulas (3.1) of Sec. 4.3 and (1.8). • 2 7
a
2N
p
l
2
2
1
p
E x a m p l e 10. p = 2, y £ %, JX2(ax
2
+ bx)dx
7
7
2 fl(2 |6| ), 2
A (a)|2«j 2
A ( )| 2
a
2 a
|-
2
1
2
X 2
1 / 2
7
( - £ ) 6(\b\ - 2 " ) , 2
7
X 2
1 / 2 X 2
(-£)fl(2 |6| ), 2
< 1,
|al 2
27
= 2,
( - £ ) 0 (2-7 T»
/
l
i
f
2
|a| 2
27
|«| 2
2 7
2
,
where tfU
27
2
1 / 2
A («)|2a|-
|n| 2
T
H>=P >
3
= 4, > 8,
( 2
3
)
Analysis on the Field oj p-Adic
Numbers
67
• Similar to the Example 9. The cases \a\ 2 "> = 2 and |n| 2 "' = 4 are considered specifically. • 2
2
2
2
The formulas (2.1) and (2.3) admit the inification. Example 11. 7 € 2 2
/
Xp
{ax +bx)dx
f P'toWK), -
\a\ p
{ \ W)l2aS*'% p
3. The Gaussian
2y
p
integrals
m *>
( - £ ) n ( - L
{
2
A
)
p
E x a m p l e 12. a / 0
j
XP(™
U«)Mp X* ( m
+ bx)dx =
2
~ S •
(3-1)
I t follows from the formula (2.3) by 7 —* 00. Note that a formula similar to (3.1) is valid in the real case Q
J X o(ax
2
0
+ bx)dx =
l/2
X ( )\2a\Z Xoo^-^y 00 a
m
= IE : (3.2)
:
trial™ = M , X » ( * ) = e x p ( - 2 i r i x ) and Aoo(a) = exp I - ' 7 signal = < V 4 / { Ifi , f
a
p
p
E
/ Js
-oo - ' 3 + M ^ l t " ' ) [ s f ^ M j , ± ) - 2 e - l « l > ] , 2
l*b > 2 | a | 3
1/2
.
(5.6 ) 6
The integrals (5.6) are calculated in a similar way to integrals (5 1) for
Analysis on ike Field oj p-Adic
Numbers
77
From the formula (5.6e) i t follows the asymptotics
K|a-*.oo .
6. Analysis
of the Function
(5.7)
S(a,q)
The function S(a, q) is denned by the formula (5.2). The function $(a,q) is entire on a, real for real a (and positive q > 0) and satisfies the functional equation e-° = S(a,q)-qS(c
xeB x$
N
B. N
Hence the equality (1.2) is valid: f(x)
=
E /(«)*e 5
W
.
Analysis
on the Field of p-Adic Numbers
81
Convergence in S we define by the following way: ft —> 0 , k —• oo in S if for any compact K C Q P
f (x)
'Mo,
h
2. Test
k^oo.
Functions
n = 1. We call a (est function every function from £ with compact support. The set of test functions is linear, we denote it by U = X>(Q ). Let ip £ T>. Then by the Lemma 1 of Sec. 6.1 there exists / € 2 , such that P
tp(x + x') = \. The following imbedding is true: N
l
V CV%„
N < N',l < I' .
N
E x a m p l e s . 1. A (x) € V , 7 € 2 . 2. 6(\x\ - p>) e VI- , 7 e Z , where y
y
1
p
-
M
ail
i
3. i ( i - k)6(\x\ - p ) £ l ? , )t = 1,2,... , p - 1, 7 € 2 , where the function 6(x(, — it) is defined by the formula (1.1). 4. I f i f is a clopen set then fljr 6 V. 5 . £ {x) p{x) G P , / = min( ,0), 7 ^ 2 . By the Lemma 2 of Sec. 6.1 every function ip from V' is represented in the form 1
0
h
7 - 1
p
X
7
7
N
*K*)=
E ik —* 0, k —> oo in V iff (i) 'Pk £ f V ° e r e N and / do not depend on fc, w
(ii) m — ' 0, k - oo. This convergence assigns the Schwartz topology in D. The space T> is complete i.e. for every convergent in itself sequence {ipk, fc —* oo), ipi, £V,
, — f \ —• 0, k,l —* oo in V, there exists a function ip G V such that ipi- —* it —» oo in T>. From the definitions it follows directly V = l i m indU^r,
V
N
=
lim mcYD'
(2.3)
N
Now we shall prove: 2?(Q ) is dense in C(K)* p
• Let / (E C(JC) and e be an arbitrary positive number. There exists a number 7 G 2 such that \f(x)-f(a)\ < e if a: G B ( a ) f l K , a G X . As the compact 7f can be covered by a finite number of disjoint discs B (a ) (see Corollary 3 from the Lemma 3 of Sec. 1.3) the characteristic functions A (x — a ) of these discs obey the property 7
v
y
v
y
£ A , ( i - 0 = l,
(2.4)
zGif,
besides A ( x — a") G V (see Example 1). Therefore 7
A(*) = £ / K ) A ( z 7
l a
')ei'
and owing to (2.4) l l / - A I | c ( K ) < max reft
" The definition of the space C ( / f ) see in Sec. 4.1
< e $ > ( * - a " ) = e. 7
Analysis on the Field of p-Adic Numbers
83
Let O be an open set in Q . The space of test functions V{0) is defined as a set of test functions from £>(Q ) — V which supports are contained in O. The space V(O) is the subspace of the space V\ its properties are similar to the properties of V as in the case of the field H (see [205]). p
P
V(0)
p
is dense in L (0),
1 < p < oo.
• I t follows from the facts that V(0) is dense in C(K) and C(K) is dence in L {0) where K is an arbitrary compact contained in O (see Sec. 4.1). p
•
In the space T>(0) the T h e o r e m o n " d e c o m p o s i t i o n o f u n i t y " is valid: let an open set 0 be a union of no more than a countable set of disjoint circles,
Then their characteristic functions composition of unity in O,
A (x yk
k
— a ),
E A ^ ( z - * ) = l, t a
xeO.
k = 1,2,... form a de-
(2.5)
In conclusion we shall prove the following L e m m a . In order thai (fy —# Q jfe —* 00 in T>, it is necessary and sufficient that the condition (i) and one of the conditions ( h i )
.
Anatyiit
on the Field of p-Adic
Numbcri
85
Therefore by the study of the space 2?' it is possible to use general theorems of the functional analysis. I n addition the theory is essentially simplified in comparison with the corresponding theory over the field K {cf. [205]): i t is sufficient to verify the linearity of functionals, their continuity follows automatically. The space V
is complete.
• Let a sequence {ft,k —* 00} of functional ft G V converge in itself, / k - / j — 0, t , / — c o i n D ' , i.e.
Hence there exists a number C( ) • k — co
(3.1)
It is clear that the functional C(p) = (f,p), f £ V, tp € T>. The equality (3.1) shows that f -> f, fc 00 in Z>'. • k
Every function / G formula (/,?)
defines a generalized function / G V = j f{x) £ V'(0') on any open set O' C G by the rule
(fo',v) = (M,
' ( Q £ ) . Note that for g = 1 the equality (6.4) is equivalent to the equality + M
3fc
k
f(x),
J p(x,y)dy l
k
= j(f(x) p(x,y))dy, }t
(A) of the following form; V(A) = P „ ( A « ) f i ( | A | ) S 2 ( | A | ) . . . , 2
2
3
3
(8.7)
where ^
(
*
)
=
e
7 b
x
p
( - y ) •
and the function f i ( | r | ) is defined in Sec. 6.1. Note, that (8.7) is the product of vacuum vectors in real and p-adic quantum mechanics (see Sec. 11). p
V I I . C o n v o l u t i o n a n d the Fourier T r a n s f o r m a t i o n In this section we study most important linear operations over generalized functions, namely the convolution and the Fourier transform operations and connected with them the multiplication operation. 1. Convolution
of Generalized
Functions
A sequence {tjt, k —> oo} of functions n £ V we call 1-sequence, i f there exists N £ Z such that k
fj ( ) t X
t
= Afcfz) = f 2 ( p - | x | ) ,
k >N .
p
It is clear that "t —* l,fc —' oo in £ . The sequence { A i , f c —» Co) we call the canonical 1-sequence. Let / and g be generalized functions from V. Their convolution we call the functional defined by the equality (f*g, v) = £ t t £ / C * 0
x
s(y), A {x)ip{x k
+ )) y
f*g
(i.i)
Analysis on the Field of p-Adie
Numbers
9S
if the limit exists for all ip G V. The right-hand side of the equality (1.1) defines an linear functional on T>, and thus f+g G V (see Sec. 6.3). Note that the equality (1.1) is equivalent to the following one (/*?,¥>) = J i m (f(x) X g(y),r, (x)(*)= £
v
c„Mx-« ) w
i', supp g C B J V then ft*9t — /*
