**AMS Chelsea Publishing**

Volume: 377;
1964;
423 pp;
Hardcover

MSC: Primary 46;

Print ISBN: 978-1-4704-2658-3

Product Code: CHEL/377.H

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-3122-8
Product Code: CHEL/377.H.E**

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

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# Generalized Functions, Volume 1: Properties and Operations

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*I. M. Gel′fand; G. E. Shilov*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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 1 is devoted to basics of the
theory of generalized functions. The first chapter contains main
definitions and most important properties of generalized functions as
functional on the space of smooth functions with compact support. The
second chapter talks about the Fourier transform of generalized
functions. In Chapter 3, definitions and properties of some important
classes of generalized functions are discussed; in particular,
generalized functions supported on submanifolds of lower dimension,
generalized functions associated with quadratic forms, and homogeneous
generalized functions are studied in detail. Many simple basic
examples make this book an excellent place for a novice to get
acquainted with the theory of generalized functions. A long appendix
presents basics of generalized functions of complex variables.

#### Readership

Graduate students and research mathematicians interested in analysis.

#### Table of Contents

# Table of Contents

## Generalized Functions, Volume 1: Properties and Operations

- Cover Cover11
- Title page iii4
- Translator’s note v6
- Foreword to the first Russian edition vii8
- Foreword to the second Russian edition xi12
- Contents xiii14
- Chapter I. Definition and simplest properties of generalized functions 120
- Chapter II. Fourier transforms of generalized functions 153172
- Chapter III. Particular types of generalized functions 209228
- Summary of fundamental definitions and equations of volume I 330349
- Table of Fourier transforms 359378
- Appendix A. Proof of the completeness of the generalized-function space 368387
- Appendix B. Generalized functions of complex variables 370389
- Notes and references to the literature 413432
- Index 419438
- Index of particular generalized functions 422441
- Back Cover Back Cover1450