傅立叶分析导论 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

傅立叶分析导论 英文版PDF格式电子书版下载

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Chapter 1．The Genesis of Fourier Analysis1

1 The vibrating string2

1.1 Derivation of the wave equation6

1.2 Solution to the wave equation8

1.3 Example: the plucked string17

2 The heat equation18

2.1 Derivation of the heat equation18

2.2 Steady-state heat equation in the disc20

3 Exercises23

4 Problem28

Chapter 2．Basic Properties of Fourier Series29

1 Examples and formulation of the problem30

1.1 Main definitions and some examples34

2 Uniqueness of Fourier series39

3 Convolutions44

4 Good kernels48

5 Cesàro and Abel summability: applications to Fourier series51

5.1 Cesàro means and summation51

5.2 Fejér's theorem52

5.3 Abel means and summation54

5.4 The Poisson kernel and Dirichlet's problem in the unit disc55

6 Exercises58

7 Problems65

Chapter 3．Convergence of Fourier Series69

1 Mean-square convergence of Fourier series70

1.1 Vector spaces and inner products70

1.2 Proof of mean-square convergence76

2 Return to pointwise convergence81

2.1 A local result81

2.2 A continuous function with diverging Fourier series83

3 Exercises87

4 Problems95

Chapter 4．Some Applications of Fourier Series100

1 The isoperimetric inequality101

2 Weyl's equidistribution theorem105

3 A continuous but nowhere differentiable function113

4 The heat equation on the circle118

5 Exercises120

6 Problems125

Chapter 5．The Fourier Transform on ？129

1 Elementary theory of the Fourier transform131

1.1 Integration of functions on the real line131

1.2 Definition of the Fourier transform134

1.3 The Schwartz space134

1.4 The Fourier transform on S136

1.5 The Fourier inversion140

1.6 The Plancherel formula142

1.7 Extension to functions of moderate decrease144

1.8 The Weierstrass approximation theorem144

2 Applications to some partial differential equations145

2.1 The time-dependent heat equation on the real line145

2.2 The steady-state heat equation in the upper half-plane149

3 The Poisson summation formula153

3.1 Theta and zeta functions155

3.2 Heat kernels156

3.3 Poisson kernels157

4 The Heisenberg uncertainty principle158

5 Exercises161

6 Problems169

Chapter 6．The Fourier Transform on ？d175

1 Preliminaries176

1.1 Symmetries176

1.2 Integration on ？d178

2 Elementary theory of the Fourier transform180

3 The wave equation in ？d × ？184

3.1 Solution in terms of Fourier transforms184

3.2 The wave equation in ？3 × ？189

3.3 The wave equation in ？2 × ？: descent194

4 Radial symmetry and Bessel functions196

5 The Radon transform and some of its applications198

5.1 The X-ray transform in ？2199

5.2 The Radon transform in ？3201

5.3 A note about plane waves207

6 Exercises207

7 Problems212

Chapter 7．Finite Fourier Analysis218

1 Fourier analysis on ？(N)219

1.1 The group ？(N)219

1.2 Fourier inversion theorem and Plancherel identity on ？(N)221

1.3 The fast Fourier transform224

2 Fourier analysis on finite abelian groups226

2.1 Abelian groups226

2.2 Characters230

2.3 The orthogonality relations232

2.4 Characters as a total family233

2.5 Fourier inversion and Plancherel formula235

3 Exercises236

4 Problems239

Chapter 8．Dirichlet's Theorem241

1 A little elementary number theory241

1.1 The fundamental theorem of arithmetic241

1.2 The infinitude of primes244

2 Dirichlet's theorem252

2.1 Fourier analysis, Dirichlet characters, and reduc-tion of the theorem254

2.2 Dirichlet L-functions255

3 Proof of the theorem258

3.1 Logarithms258

3.2 L-functions261

3.3 Non-vanishing of the L-function265

4 Exercises275

5 Problems279

Appendix: Integration281

1 Definition of the Riemann integral281

1.1 Basic properties282

1.2 Sets of measure zero and discontinuities of inte-grable functions286

2 Multiple integrals289

2.1 The Riemann integral in ？d289

2.2 Repeated integrals291

2.3 The change of variables formula292

2.4 Spherical coordinates293

3 Improper integrals. Integration over ？d294

3.1 Integration of functions of moderate decrease294

3.2 Repeated integrals295

3.3 Spherical coordinates297

Notes and References299

Bibliography301

Symbol Glossary305