Distributions and Their Applications in Physics

ForewordEditor's NoteChapter 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 OperatorsChapter 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 eChapter 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 e' Chapter 4. Local Properties of Distributions 4.1. Partitions of Unity 4.2. The Support of a DistributionChapter 5. Simple Examples of Distributions 5.1. The Dirac Measure 5.2. The Principal Value 5.3. The Sokhotski-Plemelj FormulaChapter 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 ApplicationsChapter 7. Distributions with Compact Support and the General Structure of Tempered Distributions 7.1. The Space e' 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 PointChapter 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+¿ and x-¿ 8.3.3. The Distributions 1/xn, n = 1,2,... 8.3.4. The Distributions (x±i0)¿ 8.3.5. Expansion of the Distribution-Valued Functions x±¿ in Taylor and Laurent Series 8.3.6. The Distribution r¿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 OperatorsChapter 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 e' 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 ClassChapter 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 d(P), d'(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 d±(p2-m2) and d(p2-m2) 11.5. Some Fourier TransformsChapter 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.