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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.
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. 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.
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