**AMS Chelsea Publishing**

Volume: 378;
1968;
261 pp;
Hardcover

MSC: Primary 46;

Print ISBN: 978-1-4704-2659-0

Product Code: CHEL/378.H

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-3123-5
Product Code: CHEL/378.H.E**

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

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#### Supplemental Materials

# Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions

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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 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.

#### Readership

Graduate students and research mathematicians interested in analysis and differential equations.

#### Table of Contents

# Table of Contents

## Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions

- Cover Cover11
- Title page iii4
- Preface to the Russian edition v6
- Contents ix10
- Chapter I. Linear topological spaces 112
- Chapter II. Fundamental and generalized functions 7788
- Chapter III. Fourier transformations of fundamental and generalized functions 122133
- Chapter IV. Spaces of type 𝑆 166177
- Appendix 1. Generalization of spaces of type 𝑆 244255
- Notes and references 253264
- Bibliography 257268
- Index 259270
- Back Cover Back Cover1274