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Distributions and Their Applications in Physics is the introduction of the Theory of Distributions and their applications in physics. The book contains a discussion of those topics under the Theory of Distributions that are already considered classic, which include local distributions; distributions with compact support; tempered distributions; the distribution theory in relativistic physics; and many others. The book also covers the Normed and Countably-normed Spaces; Test Function Spaces; Distribution Spaces; and the properties and operations involved in distributions. The text is recommended for physicists that wish to be acquainted with distributions and their relevance and applications as part of mathematical and theoretical physics, and for mathematicians who wish to be acquainted with the application of distributions theory for physics.
Chapter 1. Normed and Countably-Normed Spaces
1.1. Topological Spaces
1.2. Metric Spaces
1.3. Topological Linear Spaces
1.4. Normed Spaces
1.5. Countably-Normed Spaces
1.6. Continuous Linear Functionals
1.7. The Hahn-Banach Theorem
1.8. Dual Spaces, Strong and Weak Topologies on Dual Spaces
1.9. Strong and Weak Topologies on Initial Spaces
1.10. The Union and Direct Sum of Countably-Normed Spaces
1.10.1. The Union of Countably-Normed Spaces
1.10.2. The Direct Sum of Countably-Normed Spaces
1.11. Linear Operators
Chapter 2. Test Function Spaces
2.2. The Test Space D(K)
2.3. The Test Space D
2.4. The Test Space A
2.5. The Test Space ε
Chapter 3. Distribution Spaces
3.1. The Distribution Space D'(K)
3.2. The Distribution Space D'
3.3. The Distribution Space A'
3.4. The Distribution Space ε'
Chapter 4. Local Properties of Distributions
4.1. Partitions of Unity
4.2. The Support of a Distribution
Chapter 5. Simple Examples of Distributions
5.1. The Dirac Measure
5.2. The Principal Value
5.3. The Sokhotski-Plemelj Formula
Chapter 6. Operations on Distributions
6.1. Translation and Reflection
6.2. Multiplication of Distributions by Infinitely Differentiable Functions
6.3. Multiplication of Distributions
6.4. Differentiation of Distributions
6.5. Some Applications
Chapter 7. Distributions with Compact Support and the General Structure of Tempered Distributions
7.1. The Space ε' as the Space of Distributions with Compact Support
7.2. A System of Integral Norms on A
7.3. Tempered Distributions as Derivatives of Slowly Increasing Functions
7.4. The Structure of Distributions which are Concentrated at a Point
Chapter 8. Functions with Non-Integrable Algebraic Singularities
8.1. The Problem of Regularization of Divergent Integrals
8.2. Distributions which Depend on a Parameter
8.3. Regularization by Analytic Continuation
8.3.1. An Example
8.3.2. The Distributions x+λ and x-λ
8.3.3. The Distributions 1/xn, n = 1,2,...
8.3.4. The Distributions (x±i0)λ
8.3.5. Expansion of the Distribution-Valued Functions x±λ in Taylor and Laurent Series
8.3.6. The Distribution rλ
Chapter 9. The Tensor Product and the Convolution of Distributions
9.1. The Tensor Product of Distributions
9.2. The Convolution of Distributions
9.3. Regularization of Distributions
9.4. Fundamental Solutions of Linear Differential Operators
Chapter 10. Fourier Transforms
10.1. Fourier Transforms of Test Functions in A and Distributions in A'
10.2. Fourier Transforms of Test Functions in D and Distributions in D'
10.3. The Convolution Theorem
10.4. Fourier Transforms of Distributions in ε'
10.5. The Calculation of the Fourier Transforms of Certain Distributions by Analytic Continuation
10.6. A Fundamental Lemma in the Theory of Fourier-Laplace Transforms of Distributions
10.7. Fourier-Laplace Transforms of Distributions
10.8. The Product of Distributions in a Certain Class
Chapter 11. Distributions Connected with the Light Cone
11.1 Distributions Concentrated on a Smooth Surface
11.1.3. Properties of δ(P), δ'(P), ...
11.2. Distributions Concentrated on a Cone
11.3. The Solution of the Cauchy Problem for the Wave Equation
11.4. The Tempered Distributions δ±(p2-m2) and δ(p2-m2)
11.5. Some Fourier Transforms
Chapter 12. Hilbert Space and Distributions. Applications in Physics
12.1 Preliminaries: Some Elementary Remarks on Linear Operators in Hilbert Space
12.2. Analytic Vectors: Nelson's Theorem
12.3. Fock Space and the Annihilation and Creation Operators
12.3.1. Fock Space
12.3.2. The Annihilation and Creation Operators
12.3.3. Quantized Distributions
12.4. The Free Scalar Neutral Field
12.4.1. Relativistic Fock Space
12.4.2. The Free Scalar Neutral Field
12.4.3. The Two-point Function
A.1. What are Ultradistributions
A.2. Beuerling-Bjorck Ultradistributions
A.3. Fourier-Laplace Transforms
A.4. Positive Definite Ultradistributions
Other Titles in the Series
- No. of pages:
- © Pergamon 1980
- 1st January 1980
- eBook ISBN:
University of Munich, FRG
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