**AMS Chelsea Publishing**

Volume: 380;
1964;
384 pp;
Hardcover

MSC: Primary 46;

**Print ISBN: 978-1-4704-2662-0
Product Code: CHEL/380.H**

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-3125-9
Product Code: CHEL/380.H.E**

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

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# Generalized Functions, Volume 4: Applications of Harmonic Analysis

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*I. M. Gel′fand; N. Ya. Vilenkin*

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.

The main goal of Volume 4 is to develop the functional analysis
setup for the universe of generalized functions. The main notion
introduced in this volume is the notion of rigged Hilbert space (also
known as the equipped Hilbert space, or Gelfand triple). Such space
is, in fact, a triple of topological vector spaces \(E \subset H
\subset E'\), where \(H\) is a Hilbert space,
\(E'\) is dual to \(E\), and
inclusions \(E\subset H\) and \(H\subset E'\)
are nuclear operators. The book is devoted to various applications of this
notion, such as the theory of positive definite generalized functions, the theory of generalized stochastic processes, and the study of measures on linear
topological spaces.

#### Readership

Graduate students and research mathematicians interested in functional analysis.

#### Table of Contents

# Table of Contents

## Generalized Functions, Volume 4: Applications of Harmonic Analysis

- Cover Cover11
- Title page iii4
- Translator’s note v6
- Foreword vii8
- Contents ix10
- Chapter I. The kernel theorem. Nuclear spaces. Rigged Hilbert space 116
- Chapter II. Positive and positive-definite generalized functions 135150
- Chapter III. Generalized random processes 237252
- Chapter IV. Measures in linear topological spaces 303318
- Notes and references to the literature 371386
- Bibliography 377392
- Subject index 381396
- Back Cover Back Cover1402