Distributions and Their Applications in Physics (eBook)

International Series in Natural Philosophy

F. Constantinescu (Autor)

J. E. G. Farina, G. H. Fullerton (Herausgeber)

eBook Download: PDF

2017 | 1. Auflage
158 Seiten
Elsevier Science (Verlag)
978-1-4831-5020-8 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 26.7.2017
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik
• ISBN-10: 1-4831-5020-8 / 1483150208
• ISBN-13: 978-1-4831-5020-8 / 9781483150208