Distributions and Their Applications in Physics

Distributions and Their Applications in Physics

International Series in Natural Philosophy

1st Edition - January 1, 1980

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  • Author: F. Constantinescu
  • eBook ISBN: 9781483150208

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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.

Table of Contents

  • Foreword

    Editor's Note

    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

    2.1. Notation

    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

    11.1.1. Definitions

    11.1.2. Examples

    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

    Appendix. Ultradistributions

    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

Product details

  • No. of pages: 158
  • Language: English
  • Copyright: © Pergamon 1980
  • Published: January 1, 1980
  • Imprint: Pergamon
  • eBook ISBN: 9781483150208

About the Author

F. Constantinescu

Affiliations and Expertise

University of Munich, FRG

About the Editors

J. E. G. Farina

G. H. Fullerton

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