Distribution Theory

The theory of distributions has numerous applications and is extensively used in mathematics, physics and engineering. There is however relatively little elementary expository literature on distribution theory. This book is intended as an introduction. Starting with the elementary theory of distributions, it proceeds to convolution products of distributions, Fourier and Laplace transforms, tempered distributions, summable distributions and applications. The theory is illustrated by several examples, mostly beginning with the case of the real line and then followed by examples in higher dimensions. This is a justified and practical approach, it helps the reader to become familiar with the subject. A moderate number of exercises are added.

It is suitable for a one-semester course at the advanced undergraduate or beginning graduatelevelor for self-study.

Preface 2

1 Definition and first properties of distributions 7

1.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Support of a distribution . . . . . . . . . . . . . . . . . . . . . 10

2 Differentiating distributions 13

2.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 13

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 The distributions x−1+ ( 6= 0,−1,−2, . . . )* . . . . . . . . . . 16

2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Green’s formula and harmonic functions . . . . . . . . . . . . 19

2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Multiplication and convergence of distributions 27

3.1 Multiplication with a C1 function . . . . . . . . . . . . . . . 27

3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Convergence in D0 . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Distributions with compact support 31

4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 31

4.2 Distributions supported at the origin . . . . . . . . . . . . . . 32

4.3 Taylor’s formula for Rn . . . . . . . . . . . . . . . . . . . . . 33

4.4 Structure of a distribution* . . . . . . . . . . . . . . . . . . . 34

5 Convolution of distributions 36

5.1 Tensor product of distributions . . . . . . . . . . . . . . . . . 36

5.2 Convolution product of distributions . . . . . . . . . . . . . . 38

5.3 Associativity of the convolution product . . . . . . . . . . . . 44

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Newton potentials and harmonic functions . . . . . . . . . . . 45

5.6 Convolution equations . . . . . . . . . . . . . . . . . . . . . . 47

5.7 Symbolic calculus of Heaviside . . . . . . . . . . . . . . . . . 50

5.8 Volterra integral equations of the second kind . . . . . . . . . 52

5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.10 Systems of convolution equations* . . . . . . . . . . . . . . . 55

5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 The Fourier transform 57

6.1 Fourier transform of a function on R . . . . . . . . . . . . . . 57

6.2 The inversion theorem . . . . . . . . . . . . . . . . . . . . . . 60

6.3 Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . 61

6.4 Differentiability properties . . . . . . . . . . . . . . . . . . . . 62

6.5 The Schwartz space S(R) . . . . . . . . . . . . . . . . . . . . 63

6.6 The space of tempered distributions S0(R) . . . . . . . . . . . 65

6.7 Structure of a tempered distribution* . . . . . . . . . . . . . 66

6.8 Fourier transform of a tempered distribution . . . . . . . . . 67

6.9 Paley Wiener theorems on R* . . . . . . . . . . . . . . . . . . 69

6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.11 Fourier transform in Rn . . . . . . . . . . . . . . . . . . . . . 73

6.12 The heat or diffusion equation in one dimension . . . . . . . . 75

7 The Laplace transform 79

7.1 Laplace transform of a function . . . . . . . . . . . . . . . . . 79

7.2 Laplace transform of a distribution . . . . . . . . . . . . . . . 80

7.3 Laplace transform and convolution . . . . . . . . . . . . . . . 81

7.4 Inversion formula for the Laplace transform . . . . . . . . . . 84

8 Summable distributions* 87

8.1 Definition and main properties . . . . . . . . . . . . . . . . . 87

8.2 The iterated Poisson equation . . . . . . . . . . . . . . . . . . 88

8.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 89

8.4 Canonical extension of a summable distribution . . . . . . . . 91

8.5 Rank of a distribution . . . . . . . . . . . . . . . . . . . . . . 93

9 Appendix 96

9.1 The Banach-Steinhaus theorem . . . . . . . . . . . . . . . . . 96

9.2 The beta and gamma function . . . . . . . . . . . . . . . . . 103

Bibliography 108

Index 109