Properties and Operations
Published on 12. May 2014
442 pages
978-1-4832-6159-1 (ISBN)
Description
Generalized Functions, Volume 1: Properties and Operations provides a systematic development of the theory of generalized functions and problems in analysis connected with it. This book focuses on the concept of a convenient link that connects many aspects of analysis, functional analysis, theory of differential equations, representation theory of locally compact Lie groups, and theory of probability and statistics. This volume is essentially devoted to algorithmic questions of the theory, covering many applications of generalized functions to various problems of analysis. The topics discussed include the local properties of generalized functions, differentiation as a continuous operation, and Fourier Transforms of test functions. The wave equation in space of odd dimension, derivation of Green's theorem, and reducible singular points are also described. This publication is a good reference for mathematicians, researchers, and students concerned with generalized functions.
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Content
Translator's NoteForeword to the First Russian EditionForeword to the Second Russian EditionChapter I Definition and Simplest Properties of Generalized Functions 1. Test Functions and Generalized Functions 1.1. Introductory Remarks 1.2. Test Functions 1.3. Generalized Functions 1.4. Local Properties of Generalized Functions 1.5. Addition and Multiplication by a Number and by a Function 1.6. Translations, Rotations, and Other Linear Transformations on the Independent Variables 1.7. Regularization of Divergent Integrals 1.8. Convergence of Generalized Function Sequences 1.9. Complex Test Functions and Generalized Functions 1.10. Other Test-Function Spaces 2. Differentiation and Integration of Generalized Functions 2.1. Fundamental Definitions 2.2. Examples for the Case of a Single Variable 2.3. Examples for the Case of Several Variables 2.4. Differentiation as a Continuous Operation 2.5. Delta-Convergent Sequences 2.6. Differential Equations for Generalized Function 2.7. Differentiation in S 3. Regularization of Functions with Algebraic Singularities 3.1. Statement of the Problem 3.2. The Generalized Functions x+¿ and x-¿ 3.3. Even and Odd Combinations of x+¿ and x-¿ 3.4. Indefinite Integrals of x+¿ , x-¿
