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Full Description
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. Gelfand 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.
Contents
Volume 1: Definition and simplest properties of generalized functions
Fourier transforms of generalized functions
Particular types of generalized functions
Summary of fundamental definitions and equations of volume I
Table of Fourier transforms
Proof of the completeness of the generalized-function space
Generalized functions of complex variables
Notes and references to the literature
Bibliography
Index
Index of particular generalized functions
Volume 2: Linear topological spaces
Fundamental and generalized functions
Fourier transformations of fundamental and generalized functions
Spaces of type $S$
Generalization of spaces of type $S$
Notes and references
Bibliography
Index
Volume 3: Spaces of type $W$
Uniqueness classes for the Cauchy problem
Correctness classes for the Cauchy problem
Generalized eigenfunction expansions
Notes and references
Bibliography
Index
Volume 4: The kernel theorem
Nuclear spaces
Rigged Hilbert space
Positive and positive-definite generalized functions
Generalized random processes
Measures in linear topological spaces
Notes and references to the literature
Bibliography
Subject index
Volume 5: Radon transform of test functions and generalized functions on a real affine space
Integral transforms in the complex domain
Representations of the group of complex unimodular matrices in two dimensions
Harmonic analysis on the group of complex unimodular matrices in two dimensions
Integral geometry in a space of constant curvature
Harmonic analysis on spaces homogeneous with respect to the Lorentz group
Representations of the group of real unimodular matrices in two dimensions
Notes and references to the literature
Bibliography
Index
Index of Radon transforms of particular functions
Volume 6: Homogeneous spaces with a discrete stability group
Representations of the group of unimodular matrices of order 2 with elements from a locally compact topological field
Representations of adele groups
Guide to the literature
Bibliography
Index of names
Subject index
