Distribution Theory

Convolution, Fourier Transform, and Laplace Transform

AvGerrit Dijk

Inbunden, Engelska, 2013

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Beskrivning

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 graduate level or for self-study.

Produktinformation

Utgivningsdatum:2013-03-15
Mått:170 x 240 x 14 mm
Vikt:223 g
Format:Inbunden
Språk:Engelska
Serie:De Gruyter Textbook
Antal sidor:117
Upplaga:13001
Förlag:De Gruyter
ISBN:9783110295917

Utforska kategorier

Beräkning och matematisk analys inom Naturvetenskap och teknik

Mer om författaren

Gerrit van Dijk, Leiden University, The Netherlands.

Innehållsförteckning

Preface 21 Definition and first properties of distributions 71.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Support of a distribution . . . . . . . . . . . . . . . . . . . . . 102 Differentiating distributions 132.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 132.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The distributions x−1+ ( 6= 0,−1,−2, . . . )* . . . . . . . . . . 162.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Green’s formula and harmonic functions . . . . . . . . . . . . 192.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Multiplication and convergence of distributions 273.1 Multiplication with a C1 function . . . . . . . . . . . . . . . 273.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Convergence in D0 . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Distributions with compact support 314.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 314.2 Distributions supported at the origin . . . . . . . . . . . . . . 324.3 Taylor’s formula for Rn . . . . . . . . . . . . . . . . . . . . . 334.4 Structure of a distribution* . . . . . . . . . . . . . . . . . . . 345 Convolution of distributions 365.1 Tensor product of distributions . . . . . . . . . . . . . . . . . 365.2 Convolution product of distributions . . . . . . . . . . . . . . 385.3 Associativity of the convolution product . . . . . . . . . . . . 445.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 Newton potentials and harmonic functions . . . . . . . . . . . 455.6 Convolution equations . . . . . . . . . . . . . . . . . . . . . . 475.7 Symbolic calculus of Heaviside . . . . . . . . . . . . . . . . . 505.8 Volterra integral equations of the second kind . . . . . . . . . 525.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.10 Systems of convolution equations* . . . . . . . . . . . . . . . 555.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 The Fourier transform 576.1 Fourier transform of a function on R . . . . . . . . . . . . . . 576.2 The inversion theorem . . . . . . . . . . . . . . . . . . . . . . 606.3 Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . 616.4 Differentiability properties . . . . . . . . . . . . . . . . . . . . 626.5 The Schwartz space S(R) . . . . . . . . . . . . . . . . . . . . 636.6 The space of tempered distributions S0(R) . . . . . . . . . . . 656.7 Structure of a tempered distribution* . . . . . . . . . . . . . 666.8 Fourier transform of a tempered distribution . . . . . . . . . 676.9 Paley Wiener theorems on R* . . . . . . . . . . . . . . . . . . 696.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.11 Fourier transform in Rn . . . . . . . . . . . . . . . . . . . . . 736.12 The heat or diffusion equation in one dimension . . . . . . . . 757 The Laplace transform 797.1 Laplace transform of a function . . . . . . . . . . . . . . . . . 797.2 Laplace transform of a distribution . . . . . . . . . . . . . . . 807.3 Laplace transform and convolution . . . . . . . . . . . . . . . 817.4 Inversion formula for the Laplace transform . . . . . . . . . . 848 Summable distributions* 878.1 Definition and main properties . . . . . . . . . . . . . . . . . 878.2 The iterated Poisson equation . . . . . . . . . . . . . . . . . . 888.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 898.4 Canonical extension of a summable distribution . . . . . . . . 918.5 Rank of a distribution . . . . . . . . . . . . . . . . . . . . . . 939 Appendix 969.1 The Banach-Steinhaus theorem . . . . . . . . . . . . . . . . . 969.2 The beta and gamma function . . . . . . . . . . . . . . . . . 103Bibliography 108Index 109