Many physical, chemical, biological and even economic phenomena can be modeled by differential or partial differential equations and the framework of distribution theory is the most efficient way to study these equations. A solid familiarity with the language of distributions has become almost indispensable in order to treat these questions efficiently. This book presents the theory of distributions in as clear a sense as possible while providing the reader with a background containing the essential and most important results on distributions. Together with a thorough grounding, it also provides a series of exercises and detailed solutions. The Theory of Distributions is intended for master’s students in mathematics and for students preparing for the agrégation certification in mathematics or those studying the physical sciences or engineering.
Author(s): El Mustapha Ait Ben Hassi
Series: Mathematics and Statistics
Edition: 1
Publisher: Wiley/ISTE
Year: 2023
Language: English
Pages: 303
Tags: Topological Vector Spaces, Test Functions, Distributions, Fourier Transform, Fundamental Solutions
Cover
Title Page
Copyright Page
Contents
Preface
Introduction
Chapter 1. Topological Vector Spaces
1.1. Semi-norms
1.2. Topological vector space: definition and properties
1.2.1. Topology and semi-norms
1.2.2. Topological vector space
1.2.3. Locally convex topological vector space
1.2.4. Topological metrizable vector space
1.2.5. Convergence in a topological vector space
1.2.6. Applications and linear forms
1.3. Inductive limit topology
Chapter 2. Spaces of Test Functions
2.1. Multi-index notations
2.1.1. Leibniz formula
2.2. C∞ function with compact support
2.2.1. Support of a continuous function
2.2.2. Spaces of test functions
2.2.3. Convergence in D(Ω)
2.2.4. Convolution product
2.2.5. A few density results
2.3. Exercises with solutions
Chapter 3. Distributions on an Open Set of Rd
3.1. Definitions
3.1.1. Functional definition
3.1.2. Definition of order
3.1.3. Order of a distribution
3.2. Examples of distributions
3.2.1. Regular distributions
3.2.2. Non-regular distributions
3.2.3. Other examples
3.2.4. Radon measure
3.3. Convergence of sequences of distributions
3.3.1. Definition and examples
3.3.2. Other convergence results
3.4. Exercises with solutions
Chapter 4. Operations on Distributions
4.1. Multiplication by a C∞ function
4.1.1. Definition and some properties
4.1.2. Examples
4.1.3. Convergence properties
4.1.4. Solution of the equations xT = 0, xT = 1 and xT = S
4.2. Differentiation of a distribution
4.2.1. Definition and examples
4.2.2. Continuity of the differentiation operator
4.2.3. Solution of the equations T' = 0 and ∂xiT = 0
4.2.4. Jump formula in dimension 1
4.2.5. Differentiation/integration under the duality bracket
4.3. Transformations of distributions
4.3.1. Distribution translation
4.3.2. Distribution dilation
4.3.3. Distribution parity
4.3.4. Distribution homogeneity
4.4. Exercises with solutions
Chapter 5. Distribution Support
5.1. Distribution restriction and extension
5.1.1. Unit partitions
5.1.2. Distribution localization and recollement
5.2. Distribution support
5.2.1. Definition
5.2.2. Examples
5.2.3. Properties of the support
5.3. Compact support distributions
5.3.1. Definition and properties
5.3.2. Distributions with point support
5.4. Exercises with solutions
Chapter 6. Convolution of Distributions
6.1. Definition and examples
6.1.1. Convolution of two regular distributions
6.1.2. Convolution of a distribution and a function in D(Rd)
6.1.3. Density of D(Ω) in D'(Ω)
6.1.4. Convolution of two distributions
6.1.5. Some examples
6.2. Properties of convolution
6.2.1. Support of a convolution
6.2.2. Sequential continuity of the convolution product
6.2.3. Associativity and convolution
6.2.4. Differentiation and convolution
6.2.5. Translation and convolution
6.2.6. Algebraic study of D'+(R)
6.3. Exercises with solutions
Chapter 7. Schwartz Spaces and Tempered Distributions
7.1. S(Rd) Schwartz spaces
7.1.1. Definitions and examples
7.1.2. Topology and convergence in S(Rd)
7.1.3. First properties of S(Rd)
7.1.4. Operators in S(Rd)
7.2. Tempered distributions
7.2.1. Definition and examples
7.2.2. Convergence in S' (Rd)
7.2.3. First properties of S'(Rd)
7.2.4. Operators in S' (Rd)
7.3. Exercises with solutions
Chapter 8. Fourier Transform
8.1. Fourier transform in L1(Rd)
8.1.1. Definition and first properties
8.1.2. Fourier transform and operations
8.1.3. Fourier transform inversion
8.2. Fourier transform in S(Rd)
8.2.1. Definition and first properties
8.2.2. Fourier transform and operations
8.2.3. Fourier transform inversion
8.2.4. Fourier transform and convolution
8.3. Fourier transform in S'(Rd)
8.3.1. Definition and first properties
8.3.2. Fourier transform and differentiation
8.3.3. Fourier transform inversion
8.3.4. Fourier transform in E'(Rd)
8.3.5. Fourier transform and Poisson summation formula
8.3.6. Fourier transform and convolution
8.4. Exercises with solutions
Chapter 9. Applications to ODEs and PDEs
9.1. Partial Fourier transform
9.2. Tempered solutions of differential equations
9.3. Fundamental solutions of certain PDEs
9.3.1. Heat equation
9.3.2. Wave equation
Appendix
References
Index
EULA
