Theory of Distributions

Description

This book explains many fundamental ideas on the theory of distributions. The theory of partial differential equations is one of the synthetic branches of analysis that combines ideas and methods from different fields of mathematics, ranging from functional analysis and harmonic analysis to differential geometry and topology. This presents specific difficulties to those studying this field. This second edition, which consists of 10 chapters, is suitable for upper undergraduate/graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions. It can also be used for one-semester course. Preface to the Second Edition Preface to the First Edition Contents 1 Introduction 1.1 The Spaces C0∞ and S 1.2 The Lp Spaces 1.2.1 Definition 1.2.2 The Inequalities of Hölder and Minkowski 1.2.3 Some Properties 1.2.4 The Riesz–Fischer Theorem 1.2.5 Separability 1.2.6 Duality 1.2.7 General Lp Spaces 1.3 The Convolution of Locally Integrable Functions 1.4 Cones in Rn 1.5 Advanced Practical Problems 1.6 Notes and References 2 Generalities on Distributions 2.1 Definitions 2.2 Order of a Distribution 2.3 Change of Variables 2.4 Sequences and Series 2.5 Support 2.6 Singular Support 2.7 Measures 2.8 Multiplying Distributions by C∞ Functions 2.9 Advanced Practical Problems 2.10 Notes and References 3 Differentiation 3.1 Derivatives 3.2 The Local Structure of Distributions 3.3 The Primitive of a Distribution 3.4 Simple and Double Layers on Surfaces 3.5 Advanced Practical Problems 3.6 Notes and References 4 Homogeneous Distributions 4.1 Definition 4.2 Properties 4.3 Advanced Practical Problems 4.4 Notes and References 5 The Direct Product of Distributions 5.1 Definition 5.2 Properties 5.3 Advanced Practical Problems 5.4 Notes and References 6 Convolutions 6.1 Definition 6.2 Properties 6.3 Existence 6.4 The Convolution Algebras D'(Γ+) and D'(Γ) 6.5 Regularization of Distributions 6.6 Fractional Differentiation and Integration 6.7 Advanced Practical Problems 6.8 Notes and References 7 Tempered Distributions 7.1 Definition 7.2 Direct Product 7.3 Convolution 7.4 Advanced Practical Problems 7.5 Notes and References 8 Integral Transforms 8.1 The Fourier Transform in S(Rn) 8.2 The Fourier Transform in S'(Rn) 8.3 Properties of the Fourier Transform in S'(Rn) 8.4 The Fourier Transform of Distributions with Compact Support 8.5 The Fourier Transform of Convolutions 8.6 The Laplace Transform 8.6.1 Definition 8.6.2 Properties 8.7 Advanced Practical Problems 8.8 Notes and References 9 Fundamental Solutions 9.1 Definition and Properties 9.2 Fundamental Solutions of Ordinary Differential Operators 9.3 Fundamental Solution of the Heat Operator 9.4 Fundamental Solution of the Laplace Operator 9.5 Advanced Practical Problems 9.6 Notes and References 10 Sobolev Spaces 10.1 Definitions 10.2 Elementary Properties 10.3 Approximation by Smooth Functions 10.4 Extensions 10.5 Traces 10.6 Sobolev Inequalities 10.7 The Space H-s 10.8 Advanced Practical Problems 10.9 Notes and References References Index