Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions
About this Title
I. M. Gel′fand and G. E. Shilov. Translated by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer
Publication: AMS Chelsea Publishing
Publication Year: 1968; Volume 378
ISBNs: 978-1-4704-2659-0 (print); 978-1-4704-3123-5 (online)
The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory.
Volume 2 is devoted to detailed study of generalized functions as linear functionals on appropriate spaces of smooth test functions. In Chapter 1, the authors introduce and study countable-normed linear topological spaces, laying out a general theoretical foundation for the analysis of spaces of generalized functions. The two most important classes of spaces of test functions are spaces of compactly supported functions and Schwartz spaces of rapidly decreasing functions. In Chapters 2 and 3 of the book, the authors transfer many results presented in Volume 1 to generalized functions corresponding to these more general spaces. Finally, Chapter 4 is devoted to the study of the Fourier transform; in particular, it includes appropriate versions of the Paley–Wiener theorem.
Graduate students and research mathematicians interested in analysis and differential equations.
Table of Contents
- Chapter I. Linear topological spaces
- Chapter II. Fundamental and generalized functions
- Chapter III. Fourier transformations of fundamental and generalized functions
- Chapter IV. Spaces of type $S$
- Appendix 1. Generalization of spaces of type $S$