El Mustapha Ait Ben Hassi 
The Theory of Distributions 
Introduction

Many physical, chemical, biological and even economic phenomena can be modeled by differential or partial differential equations, and the framework of distribution theory is the most efficient way to study these equations. A solid familiarity with the language of distributions has become almost indispensable in order to treat these questions efficiently.

This book presents the theory of distributions in as clear a sense as possible while providing the reader with a background containing the essential and most important results on distributions. Together with a thorough grounding, it also provides a series of exercises and detailed solutions.

The Theory of Distributions is intended for master’s students in mathematics and for students preparing for the agrégation certification in mathematics or those studying the physical sciences or engineering.

قائمة المحتويات

Preface ix

Introduction xi

Chapter 1 Topological Vector Spaces 1

1.1 Semi-norms 1

1.2 Topological vector space: definition and properties 2

1.3 Inductive limit topology 9

Chapter 2 Spaces of Test Functions 13

2.1 Multi-index notations 13

2.2 C infinity function with compact support 14

2.3 Exercises with solutions 26

Chapter 3 Distributions on an Open Set of Rd 37

3.1 Definitions 37

3.2 Examples of distributions 39

3.3 Convergence of sequences of distributions 48

3.4 Exercises with solutions 55

Chapter 4 Operations on Distributions 75

4.1 Multiplication by a C infinity function 75

4.2 Differentiation of a distribution 81

4.3 Transformations of distributions 100

4.4 Exercises with solutions 103

Chapter 5 Distribution Support 123

5.1 Distribution restriction and extension 123

5.2 Distribution support 126

5.3 Compact support distributions 132

5.4 Exercises with solutions 137

Chapter 6 Convolution of Distributions 151

6.1 Definition and examples 151

6.2 Properties of convolution 161

6.3 Exercises with solutions 167

Chapter 7 Schwartz Spaces and Tempered Distributions 179

7.1 S(Rd) Schwartz spaces 179

7.2 Tempered distributions 189

7.3 Exercises with solutions 196

Chapter 8 Fourier Transform 205

8.1 Fourier transform in L1(Rd) 205

8.2 Fourier transform in S(Rd) 220

8.2.1 Definition and first properties 220

8.3 Fourier transform in S(Rd) 230

8.4 Exercises with solutions 240

Chapter 9 Applications to ODEs and PDEs 263

9.1 Partial Fourier transform 263

9.2 Tempered solutions of differential equations 264

9.3 Fundamental solutions of certain PDEs 265

Appendix 269

References 275

Index 277