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.
Author(s): Svetlin G. Georgiev
Edition: 2
Publisher: Springer Nature Switzerland AG
Year: 2021
Language: English
Pages: 262
City: Cham
Tags: Distributions, PDE, Fourier Transform, Laplace Transform, Fundamental Solutions, Sobolev Spaces
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
