Constantinescu / Farina / Fullerton Distributions and Their Applications in Physics

International Series in Natural Philosophy
1. Auflage 2017
ISBN: 978-1-4831-5020-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 158 Seiten, Web PDF

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.

Autoren/Hrsg.

Constantinescu, F.

Farina, J. E. G.

Fullerton, G. H.

Inhaltsverzeichnis

1;Front Cover;1
2;Distributions and their Applications in Physics;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;10
6;Editor's Note;11
7;CHAPTER 1. Normed and Countably-normed Spaces;12
7.1;1.1. Topological Spaces;12
7.2;1.2. Metric Spaces;14
7.3;1.3. Topological Linear Spaces;14
7.4;1.4. Normed Spaces;15
7.5;1.5. Countably-Normed Spaces;17
7.6;1.6. Continuous Linear Functionals;20
7.7;1.7. The Hahn-Banach Theorem;22
7.8;1.8. Dual Spaces, Strong and Weak Topologies on Dual Spaces;23
7.9;1.9. Strong and Weak Topologies on Initial Spaces;29
7.10;1.10. The Union and Direct Sum of Countably-Normed Spaces;31
7.11;1.11. Linear Operators;33
8;CHAPTER 2. Test Function Spaces;35
8.1;2.1· Notation;35
8.2;2.2. The Test Space D(.);36
8.3;2.3. The Test Space D;37
8.4;2.4. The Test Space .;38
8.5;2.5. The Test Space .;41
9;CHAPTER 3. Distribution Spaces;42
9.1;3.1. The Distribution Space D' (.);42
9.2;3.2. The Distribution Space D';43
9.3;3.3· The Distribution Space .';43
9.4;3·4. The Distribution Space .';44
10;CHAPTER 4. Local Properties of Distributions;45
10.1;4·1. Partitions of Unity;45
10.2;4.2. The Support of a Distribution;49
11;CHAPTER 5. Simple Examples of Distributions;51
11.1;5.1. The Dirac Measure;51
11.2;5.2. The Principal Value;52
12;CHAPTER 6. Operations on Distributions;54
12.1;6.1. Translation and Reflection;54
12.2;6.2. Multiplication of Distributions by Infinitely Differentiable Functions;55
12.3;6.3. The Multiplication of Distributions;56
12.4;6.4. Differentiation of Distributions;56
12.5;6.5. Some Applications;57
13;CHAPTER 7. Distributions with Compact Support and the General Structure of Tempered Distributions;60
13.1;7.1. The space .' as the Space of Distributions with Compact Support;60
13.2;7.2. A System of Integral Norms on A;61
13.3;7.3. Tempered Distributions as Derivatives of Slowly Increasing Functions;62
13.4;7.4. The Structure of Distributions which are Concentrated at a Point;64
14;CHAPTER 8. Functions with Non-integrable Algebraic Singularities;67
14.1;8.1. The Problem of Regularization of Divergent Integrals;67
14.2;8.2. Distributions which Depend on a Parameter;68
14.3;8.3. Regularization by Analytic Continuation;72
14.4;CHAPTER 9. The Tensor Product and the Convolution of Distributions;82
14.5;9.1. The Tensor Product of Distributions;82
14.6;9.2. The Convolution of Distributions;86
14.7;9.3. Regularization of Distributions;88
14.8;9.4. Fundamental Solutions of Linear Differential Operators;89
15;CHAPTER 10. Fourier Transforms;91
15.1;10.1. Fourier Transforms of Test Functions in . and Distributions in .';91
15.2;10.2. Fourier Transforms of Test Function in D and Distributions in D';94
15.3;10.3· The Convolution Theorem;96
15.4;10.4. Fourier Transforms of Distributions in .';97
15.5;10.5. The Calculation of the Fourier Transforms of Certain Distributions by Analytic Continuation;100
15.6;10.6. A Fundamental Lemma in the Theory of Fourier-Laplace Transforms of Distributions;103
15.7;10.7. Fourier-Laplace Transforms of Distributions;105
15.8;10.8. The Product of Distributions in a Certain Class;111
16;CHAPTER 11. Distributions Connected with the Light Cone;112
16.1;11.1. Distributions which are Concentrated in a Smooth Surface;112
16.2;11.2· Distributions Concentrated on a Cone;117
16.3;11·3· The Solution of the Cauchy Problem for the Wave Equation;121
16.4;11.4· The Tempered Distributions;123
16.5;11·5· Some Fourier Transforms;126
17;CHAPTER 12. Hilbert Space and Distributions. Applications in Physics;128
17.1;12·1· Preliminaries : Some Elementary Remarks on Linear Operators in Hilbert Space;128
17.2;12·2· Analytic Vectors: Nelson's Theorem;133
17.3;12.3· Fock Space and the Annihilation and Creation Operators;135
17.4;12.4· The Free Scalar Neutral Field;141
18;Appendix Ultradistributions;146
18.1;.·1· What are Ultradistributions;146
18.2;A.2. Beuerling-Bjorck Ultradistributions;146
18.3;A.3· Fourier-Laplace Transforms;149
18.4;A.4. Positive definite Ultradistributions;151
19;References;153
20;Index;156
21;OTHER TITLES IN THE SERIES IN NATURAL PHILOSOPHY;158