Distribution Theory (eBook)

Convolution, Fourier Transform, and Laplace Transform

Gerrit Dijk (Autor)

eBook Download: PDF

2013
117 Seiten
De Gruyter (Verlag)
978-3-11-029851-2 (ISBN)

Infos

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.

Gerrit van Dijk, Leiden University, The Netherlands.

Autor

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Gerrit van Dijk, Leiden University, The Netherlands.

Inhalt

Preface 5
1 Introduction 9
2 Definition and First Properties of Distributions 11
2.1 Test Functions 11
2.2 Distributions 12
2.3 Support of a Distribution 14
3 Differentiating Distributions 17
3.1 Definition and Properties 17
3.2 Examples 18
3.3 The Distributions x+?-1(??0,-1,-2,...)* 20
3.4 Exercises 22
3.5 Green’s Formula and Harmonic Functions 22
3.6 Exercises 28
4 Multiplication and Convergence of Distributions 30
4.1 Multiplication with a C8 Function 30
4.2 Exercises 31
4.3 Convergence in D' 31
4.4 Exercises 32
5 Distributions with Compact Support 34
5.1 Definition and Properties 34
5.2 Distributions Supported at the Origin 35
5.3 Taylor’s Formula for Rn 35
5.4 Structure of a Distribution* 36
6 Convolution of Distributions 39
6.1 Tensor Product of Distributions 39
6.2 Convolution Product of Distributions 41
6.3 Associativity of the Convolution Product 47
6.4 Exercises 47
6.5 Newton Potentials and Harmonic Functions 48
6.6 Convolution Equations 50
6.7 Symbolic Calculus of Heaviside 53
6.8 Volterra Integral Equations of the Second Kind 55
6.9 Exercises 57
6.10 Systems of Convolution Equations* 58
6.11 Exercises 59
7 The Fourier Transform 60
7.1 Fourier Transform of a Function on R 60
7.2 The Inversion Theorem 62
7.3 Plancherel’s Theorem 64
7.4 Differentiability Properties 65
7.5 The Schwartz Space S(R) 66
7.6 The Space of Tempered Distributions S'(R) 68
7.7 Structure of a Tempered Distribution* 69
7.8 Fourier Transform of a Tempered Distribution 71
7.9 Paley Wiener Theorems on R* 73
7.10 Exercises 76
7.11 Fourier Transform in Rn 77
7.12 The Heat or Diffusion Equation in One Dimension 79
8 The Laplace Transform 82
8.1 Laplace Transform of a Function 82
8.2 Laplace Transform of a Distribution 83
8.3 Laplace Transform and Convolution 84
8.4 Inversion Formula for the Laplace Transform 87
9 Summable Distributions* 90
9.1 Definition and Main Properties 90
9.2 The Iterated Poisson Equation 91
9.3 Proof of the Main Theorem 92
9.4 Canonical Extension of a Summable Distribution 93
9.5 Rank of a Distribution 95
10 Appendix 98
10.1 The Banach Steinhaus Theorem 98
10.2 The Beta and Gamma Function 105
11 Hints to the Exercises 110
References 115
Index 117

Produkt-Details

• Erscheint lt. Verlag: 22.3.2013
• Reihe/Serie: De Gruyter Textbook
• Zusatzinfo: 1 b/w ill.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Distibution Theory • distributions • Distribution Theory • Fourier transform • generalized functions • heat equation • Laplace transform • Tempered Distribution
• ISBN-10: 3-11-029851-1 / 3110298511
• ISBN-13: 978-3-11-029851-2 / 9783110298512