**AMS Chelsea Publishing**

Volume: 381;
1966;
449 pp;
Hardcover

MSC: Primary 22;

Print ISBN: 978-1-4704-2663-7

Product Code: CHEL/381.H

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-3126-6
Product Code: CHEL/381.H.E**

List Price: $50.00

AMS Member Price: $45.00

MAA Member Price: $45.00

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# Generalized Functions, Volume 5: Integral Geometry and Representation Theory

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*I. M. Gel′fand; M. I. Graev; 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 unifying idea of Volume 5 in the series is the application of
the theory of generalized functions developed in earlier volumes to
problems of integral geometry, to representations of Lie groups,
specifically of the Lorentz group, and to harmonic analysis on
corresponding homogeneous spaces. The book is written with great
clarity and requires little in the way of special previous knowledge
of either group representation theory or integral geometry; it is also
independent of the earlier volumes in the series. The exposition
starts with the definition, properties, and main results related to
the classical Radon transform, passing to integral geometry in complex
space, representations of the group of complex unimodular matrices of
second order, and harmonic analysis on this group and on most
important homogeneous spaces related to this group. The volume ends
with the study of representations of the group of real unimodular
matrices of order two.

#### Readership

Graduate students and research mathematicians interested in integral geometry and representation theory.

#### Table of Contents

# Table of Contents

## Generalized Functions, Volume 5: Integral Geometry and Representation Theory

- Cover Cover11
- Title page iii4
- Translator’s note v6
- Foreword vii8
- Contents ix10
- Chapter I. Radon transform of test functions and generalized functions on a real affine space 120
- Chapter II. Integral transforms in the complex domain 7594
- Chapter III. Representations of the group of complex unimodular matrices in two dimensions 133152
- Chapter IV. Harmonic analysis on the group of complex unimodular matrices in two dimensions 202221
- Chapter V. Integral geometry in a space of constant curvature 273292
- Chapter VI. Harmonic analysis on spaces homogeneous with respect to the Lorentz group 331350
- Chapter VII. Representations of the group of real unimodular matrices in two dimensions 390409
- Notes and references to the literature 440459
- Bibliography 442461
- Index 445464
- Index of Radon transforms of particular functions 449468
- Back Cover Back Cover1474