Distribution Theory: Convolution, Fourier Transform, and Laplace Transform

Description

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. Every chapter starts with a summary and learning targets Includes manyexcercises Cocise and well-structured introduction to the topic Preface 1 Introduction 2 Definition and First Properties of Distributions 2.1 Test Functions 2.2 Distributions 2.3 Support of a Distribution 3 Differentiating Distributions 3.1 Definition and Properties 3.2 Examples 3.3 The Distributions x+?-1(??0,-1,-2,...)* 3.4 Exercises 3.5 Green’s Formula and Harmonic Functions 3.6 Exercises 4 Multiplication and Convergence of Distributions 4.1 Multiplication with a C8 Function 4.2 Exercises 4.3 Convergence in D' 4.4 Exercises 5 Distributions with Compact Support 5.1 Definition and Properties 5.2 Distributions Supported at the Origin 5.3 Taylor’s Formula for Rn 5.4 Structure of a Distribution* 6 Convolution of Distributions 6.1 Tensor Product of Distributions 6.2 Convolution Product of Distributions 6.3 Associativity of the Convolution Product 6.4 Exercises 6.5 Newton Potentials and Harmonic Functions 6.6 Convolution Equations 6.7 Symbolic Calculus of Heaviside 6.8 Volterra Integral Equations of the Second Kind 6.9 Exercises 6.10 Systems of Convolution Equations* 6.11 Exercises 7 The Fourier Transform 7.1 Fourier Transform of a Function on R 7.2 The Inversion Theorem 7.3 Plancherel’s Theorem 7.4 Differentiability Properties 7.5 The Schwartz Space S(R) 7.6 The Space of Tempered Distributions S'(R) 7.7 Structure of a Tempered Distribution* 7.8 Fourier Transform of a Tempered Distribution 7.9 Paley Wiener Theorems on R* 7.10 Exercises 7.11 Fourier Transform in Rn 7.12 The Heat or Diffusion Equation in One Dimension 8 The Laplace Transform 8.1 Laplace Transform of a Function 8.2 Laplace Transform of a Distribution 8.3 Laplace Transform and Convolution 8.4 Inversion Formula for the Laplace Transform 9 Summable Distributions* 9.1 Definition and Main Properties 9.2 The Iterated Poisson Equation 9.3 Proof of the Main Theorem 9.4 Canonical Extension of a Summable Distribution 9.5 Rank of a Distribution 10 Appendix 10.1 The Banach Steinhaus Theorem 10.2 The Beta and Gamma Function 11 Hints to the Exercises References Index