一本专家级的书 Theory of p-adic Distributions- Linear and Nonlinear Models
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Theory of p-adic Distributions:
Linear and Nonlinear Models
S. ALBEVERIO
Institut f¨ur Angewandte Mathematik, Universit¨at Bonn,
Endenicherallee 60, D-53115 Bonn, Germany;
H. C. M., I.Z.K.S., SFB 611; BiBoS.
A. YU. KHRENNIKOV
International Center for Mathematical Modeling in Physics
and Cognitive Sciences MSI, V¨axj¨o University,
SE-351 95, V¨axj¨o, Sweden.
V. M. SHELKOVICH
Department of Mathematics, St. Petersburg State Architecture and
Civil Engineering University, 2-ja Krasnoarmeiskaya 4,
190005, St. Petersburg, Russia.
cambridge university press
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C

S. Albeverio, A. Yu. Khrennikov and V. M. Shelkovich 2010
This publication is in copyright. Subject to statutory exception
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no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2010
Printed in the United Kingdom at the University Press, Cambridge
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ISBN 978-0-521-14856-6 Paperback
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To Solvejg, Irina, Evgenia
Contents
Preface page xi
1 p-adic numbers 1
1.1 Introduction 1
1.2 Archimedean and non-Archimedean normed fields 1
1.3 Metrics and norms on the field of rational numbers 6
1.4 Construction of the completion of a normed field 10
1.5 Construction of the field of p-adic numbers Qp 14
1.6 Canonical expansion of p-adic numbers 15
1.7 The ring of p-adic integers Zp 19
1.8 Non-Archimedean topology of the field Qp 21
1.9 Qp in connection with R 25
1.10 The space Qn
p 33
2 p-adic functions 35
2.1 Introduction 35
2.2 p-adic power series 35
2.3 Additive and multiplicative characters of the field Qp 40
3 p-adic integration theory 47
3.1 Introduction 47
3.2 The Haar measure and integrals 47
3.3 Some simple integrals 51
3.4 Change of variables 52
4 p-adic distributions 54
4.1 Introduction 54
4.2 Locally constant functions 54
4.3 The Bruhat.Schwartz test functions 56
4.4 The Bruhat.Schwartz distributions (generalized functions) 58
4.5 The direct product of distributions 63
vii
viii Contents
4.6 The Schwartz “kernel” theorem 64
4.7 The convolution of distributions 65
4.8 The Fourier transform of test functions 68
4.9 The Fourier transform of distributions 71
5 Some results from p-adic L1- and L2-theories 75
5.1 Introduction 75
5.2 L1-theory 75
5.3 L2-theory 77
6 The theory of associated and quasi associated homogeneous
p-adic distributions 80
6.1 Introduction 80
6.2 p-adic homogeneous distributions 80
6.3 p-adic quasi associated homogeneous distributions 83
6.4 The Fourier transform of p-adic quasi associated homogeneous
distributions 93
6.5 New type of p-adic
-functions 94
7 p-adic Lizorkin spaces of test functions and distributions 97
7.1 Introduction 97
7.2 The real case of Lizorkin spaces 98
7.3 p-adic Lizorkin spaces 99
7.4 Density of the Lizorkin spaces of test functions in Lヱ(Qn
p) 102
8 The theory of p-adic wavelets 106
8.1 Introduction 106
8.2 p-adic Haar type wavelet basis via the real Haar wavelet basis 111
8.3 p-adic multiresolution analysis (one-dimensional case) 112
8.4 Construction of the p-adic Haar multiresolution analysis 115
8.5 Description of one-dimensional 2-adic Haar wavelet bases 121
8.6 Description of one-dimensional p-adic Haar wavelet bases 128
8.7 p-adic refinable functions and multiresolution analysis 140
8.8 p-adic separable multidimensional MRA 149
8.9 Multidimensional p-adic Haar wavelet bases 151
8.10 One non-Haar wavelet basis in L2(Qp) 155
8.11 One infinite family of non-Haar wavelet bases in L2(Qp) 161
8.12 Multidimensional non-Haar p-adic wavelets 166
8.13 The p-adic Shannon.Kotelnikov theorem 168
8.14 p-adic Lizorkin spaces and wavelets 170
9 Pseudo-differential operators on the p-adic Lizorkin spaces 173
9.1 Introduction 173
9.2 p-adic multidimensional fractional operators 175
9.3 A class of pseudo-differential operators 182
Contents ix
9.4 Spectral theory of pseudo-differential operators 184
10 Pseudo-differential equations 193
10.1 Introduction 193
10.2 Simplest pseudo-differential equations 194
10.3 Linear evolutionary pseudo-differential equations of the first
order in time 197
10.4 Linear evolutionary pseudo-differential equations of the second
order in time 202
10.5 Semi-linear evolutionary pseudo-differential equations 205
11 A p-adic Schr¨odinger-type operator with point interactions 209
11.1 Introduction 209
11.2 The equation Dメ . ルI = ヤx 210
11.3 Definition of operator realizations of Dメ + V in L2(Qp) 216
11.4 Description of operator realizations 218
11.5 Spectral properties 219
11.6 The case of ョ-self-adjoint operator realizations 221
11.7 The Friedrichs extension 222
11.8 Two points interaction 224
11.9 One point interaction 226
12 Distributional asymptotics and p-adic Tauberian theorems 230
12.1 Introduction 230
12.2 Distributional asymptotics 231
12.3 p-adic distributional quasi-asymptotics 231
12.4 Tauberian theorems with respect to asymptotics 234
12.5 Tauberian theorems with respect to quasi-asymptotics 240
13 Asymptotics of the p-adic singular Fourier integrals 247
13.1 Introduction 247
13.2 Asymptotics of singular Fourier integrals for the real case 249
13.3 p-adic distributional asymptotic expansions 250
13.4 Asymptotics of singular Fourier integrals (ヰ1(x) ≌ 1) 251
13.5 Asymptotics of singular Fourier integrals (ヰ1(x) ≌ 1) 259
13.6 p-adic version of the Erdⅴelyi lemma 261
14 Nonlinear theories of p-adic generalized functions 262
14.1 Introduction 262
14.2 Nonlinear theories of distributions (the real case) 264
14.3 Construction of the p-adic Colombeau.Egorov algebra 270
14.4 Properties of Colombeau.Egorov generalized functions 272
14.5 Fractional operators in the Colombeau.Egorov algebra 276
14.6 The algebra A. of p-adic asymptotic distributions 278
14.7 A. as a subalgebra of the Colombeau.Egorov algebra 284
x Contents
A The theory of associated and quasi associated
homogeneous real distributions 285
A.1 Introduction 285
A.2 Definitions of associated homogeneous distributions and their
analysis 287
A.3 Symmetry of the class of distributions AH0(R) 295
A.4 Real quasi associated homogeneous distributions 298
A.5 Real multidimensional quasi associated homogeneous
distributions 308
A.6 The Fourier transform of real quasi associated homogeneous
distributions 313
A.7 New type of real
-functions 314
B Two identities 317
C Proof of a theorem on weak asymptotic expansions 319
D One “natural” way to introduce a measure on Qp 331
References 333
Index