This 2010 book was the first devoted to the theory of p-adic wavelets and pseudo-differential equations in the framework of distribution theory. This relatively recent theory has become increasingly important in the last decade with exciting applications in a variety of fields, including biology, image analysis, psychology, and information science. p-Adic mathematical physics also plays an important role in quantum mechanics and quantum field theory, the theory of strings, quantum gravity and cosmology, and solid state physics. The authors include many new results, some of which constitute new areas in p-adic analysis related to the theory of distributions, such as wavelet theory, the theory of pseudo-differential operators and equations, asymptotic methods, and harmonic analysis. Any researcher working with applications of p-adic analysis will find much of interest in this book. Its extended introduction and self-contained presentation also make it accessible to graduate students approaching the theory for the first time.
Author(s): S. Albeverio, A. Yu Khrennikov, V. M. Shelkovich
Series: London Mathematical Society Lecture Note Series
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 370
Tags: Математика;Математическая физика;
Cover......Page 1
Title......Page 5
Copyright......Page 6
Decaition......Page 7
Contents......Page 9
Preface......Page 13
1.2 Archimedean and non-Archimedean normed fields......Page 19
1.3.1 p-adic norm......Page 24
1.3.2 The Ostrovski theorem......Page 26
1.4 Construction of the completion of a normed field......Page 28
1.5 Construction of the field of p-adic numbers Qp......Page 32
1.6 Canonical expansion of p-adic numbers......Page 33
1.7 The ring of p-adic integers Zp......Page 37
1.8 Non-Archimedean topology of the field Qp......Page 39
1.9.1 Cantor-like sets......Page 43
1.9.2 Zp and the Cantor-like sets......Page 45
1.9.3 Qp and the Cantor-like sets......Page 46
1.9.4 Qp and the Monna map......Page 48
1.10 The space Qpn......Page 51
2.2.1 p-adic sequences......Page 53
2.2.2 p-adic series......Page 54
2.2.3 Analytic functions......Page 57
2.3.1 Additive characters of the field Qp......Page 58
2.3.2 Multiplicative characters of the field Qp......Page 61
3.2 The Haar measure and integrals......Page 65
3.3 Some simple integrals......Page 69
3.4 Change of variables......Page 70
4.2 Locally constant functions......Page 72
4.3 The Bruhat-Schwartz test functions......Page 74
4.4.1 The space D'(Qpn)......Page 76
4.4.3 The Dirac delta-function......Page 78
4.4.4 Theorem of ''piecewise sewing''......Page 79
4.4.5 Linear operators in D'(Qpn)......Page 80
4.5 The direct product of distributions......Page 81
4.6 The Schwartz ''kernel'' theorem......Page 82
4.7 The convolution of distributions......Page 83
4.8 The Fourier transform of test functions......Page 86
4.9 The Fourier transform of distributions......Page 89
5.2 L1-theory......Page 93
5.3 L2-theory......Page 95
6.2.1 Definition and characterization......Page 98
6.2.2 The Fourier transform......Page 100
6.3.1 Definition and characterization......Page 101
6.3.2 Multidimensional case......Page 110
6.4 The Fourier transform of p-adic quasi associated homogeneous distributions......Page 111
6.5 New type of p-adic Γ-functions......Page 112
7.1 Introduction......Page 115
7.2 The real case of Lizorkin spaces......Page 116
7.3.1 Lizorkin space of the first kind......Page 117
7.3.2 Lizorkin space of the second kind......Page 119
7.4 Density of the Lizorkin spaces of test functions in L(Qpn)......Page 120
8.1 Introduction......Page 124
8.2 p-adic Haar type wavelet basis via the real Haar wavelet basis......Page 129
8.3.1 Definition of the p-adic multiresolution analysis......Page 130
8.3.2 p-adic refinement equation......Page 131
8.4.1 Construction of the p-adic MRA generated by the refinable function (x)=(|x|p)to.......Page 133
8.4.2 The Haar wavelet basis......Page 134
8.5.1 Wavelet functions......Page 139
8.5.2 Real wavelet functions......Page 144
8.6 Description of one-dimensional p-adic Haar wavelet bases......Page 146
8.7.1 Construction of refinable functions......Page 158
8.7.2 Refinable functions and wavelet bases......Page 165
8.8 p-adic separable multidimensional MRA......Page 167
8.9.1 Description of multidimensional 2-adic Haar MRA......Page 169
8.9.2 One multidimensional p-adic Haar basis......Page 171
8.9.3 Multidimensional p-adic Haar wavelet bases......Page 172
8.10 One non-Haar wavelet basis in L2(Qp)......Page 173
8.11 One infinite family of non-Haar wavelet bases in L2(Qp)......Page 179
8.12 Multidimensional non-Haar p-adic wavelets......Page 184
8.13 The p-adic Shannon-Kotelnikov theorem......Page 186
8.14 p-adic Lizorkin spaces and wavelets......Page 188
9.1 Introduction......Page 191
9.2.1 The Vladimirov operator......Page 193
9.2.2 The Taibleson operator......Page 197
9.2.3 p-adic Laplacians......Page 199
9.3.1 Pseudo-differential operators......Page 200
9.4.1 Spectral analysis as wavelet analysis......Page 202
9.4.2 Wavelets as eigenfunctions of the Taibleson fractional operator......Page 207
9.4.3 Compactly supported eigenfunctions of pseudo-differential operators......Page 208
10.1 Introduction......Page 211
10.2 Simplest pseudo-differential equations......Page 212
10.3 Linear evolutionary pseudo-differential equations of the first order in time......Page 215
10.4 Linear evolutionary pseudo-differential equations of the second order in time......Page 220
10.5 Semi-linear evolutionary pseudo-differential equations......Page 223
11.1 Introduction......Page 227
11.2 The equation Dα - λI = δx......Page 228
11.3 Definition of operator realizations of Dα + V in L2(Qp)......Page 234
11.4 Description of operator realizations......Page 236
11.5 Spectral properties......Page 237
11.6 The case of n-self-adjoint operator realizations......Page 239
11.7 The Friedrichs extension......Page 240
11.8.1 Invariance with respect to the change of points of interaction......Page 242
11.8.2 Examples of P-self-adjoint realizations......Page 243
11.9 One point interaction......Page 244
12.1 Introduction......Page 248
12.3 p-adic distributional quasi-asymptotics......Page 249
12.4 Tauberian theorems with respect to asymptotics......Page 252
12.5 Tauberian theorems with respect to quasi-asymptotics......Page 258
13.1 Introduction......Page 265
13.2 Asymptotics of singular Fourier integrals for the real case......Page 267
13.3 p-adic distributional asymptotic expansions......Page 268
13.4.1 The case fπα;m(x) = |x|αp-1logmp|x|p......Page 269
13.4.2 The case f0m(x) = P(log pm|x| p|x| p), m = 0,1,2,. . .......Page 272
13.5 Asymptotics of singular Fourier integrals (π1(x) ≢ 1)......Page 277
13.6 p-adic version of the Erdelyi lemma......Page 279
14.1 Introduction......Page 280
14.2.1 Colombeau type algebras......Page 282
14.2.2 Algebras of asymptotic distributions......Page 284
14.3.1 Regularization of distributions......Page 288
14.3.2 The basic construction......Page 289
14.4.2 Associated distribution......Page 290
14.4.3 The point value......Page 291
14.4.5 Convolution......Page 293
14.5 Fractional operators in the Colombeau-Egorov algebra......Page 294
14.6.1 Weak asymptotics of regularizations of distributions......Page 296
14.6.2 Asymptotic distributions......Page 298
14.6.3 A* as a convolution algebra.......Page 301
14.7 A* as a subalgebra of the Colombeau-Egorov algebra......Page 302
A.1 Introduction......Page 303
A.2.1 Associated homogeneous distributions......Page 305
A.2.2 Analysis of definitions of AHD......Page 309
A.3 Symmetry of the class of distributions AH0(R)......Page 313
A.4.1 A class of distributions AH1(R)......Page 316
A.4.2 QAHDs......Page 320
A.5 Real multidimensional quasi associated homogeneous distributions......Page 326
A.6 The Fourier transform of real quasi associated homogeneous distributions......Page 331
A.7 New type of real Γ -functions......Page 332
B Two identities......Page 335
C Proof of a theorem on weak asymptotic expansions......Page 337
D One ``natural'' way to introduce a measure on Qp......Page 349
References......Page 351
Index......Page 366
