Table of Contents

Preface 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 Q_p 14

1.6 Canonical expansion of p-adic numbers 15

1.7 The ring of p-adic integers Z_p 19

1.8 Non-Archimedean topology of the field Q_p 21

1.9 Q_p 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 Q_p 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

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 Γ-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^p(Q_np) 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(Q_p) 155

8.11 One infinite family of non-Haar wavelet bases in L²(Q_p) 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 Lizorldn spaces 173

9.1 Introduction 173

9.2 p-adic multidimensional fractional operators 175

9.3 A class of pseudo-differential operators 182

9.4 Spectral theory of pseudo-differential operators 184

10 Pseudo-differential equations 193

10.1 Introduction 93

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 Schrödinger-type operator with point interactions 209

11.1 Introduction 209

11.2 The equation D^α - λI = δ_x 210

11.3 Definition of operator realizations of D^α + V in L₂(Q_p) 216

11.4 Description of operator realizations 218

11.5 Spectral properties 219

11.6 The case of η-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 (π₁(x) ≡ 1) 251

13.5 Asymptotics of singular Fourier integrals (π₁(x) ≠ 1) 259

13.6 p-adic version of the Erdélyi 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

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 Γ-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 Q^p 331

References 333

Index 348