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Generalized functions, volume 2 : spaces of fundamental and generalized functions.

By: Gelʹfand, I. M. (Izrailʹ Moiseevich)Contributor(s): Shilov, G. E. (Georgiĭ Evgenʹevich)Material type: TextTextLanguage: English Publication details: Providence, Rhode Island : American Mathematical Society Chelsea Publishing, 2016. Description: x, 261p. : illustrations (black and white) ; 26 cmISBN: 9781470426590; 9781470428853Uniform titles: Obobshchennye funkt︠s︡ii. English Subject(s): Theory of distributions (Functional analysis) | Functional analysis -- Distributions, generalized functions, distribution spaces -- Distributions, generalized functions, distribution spaces | Functional analysis -- Numerical solutions | Functional equations -- Numerical solutions | Mathematics -- Calculus | Mathematics -- Mathematical AnalysisDDC classification: 517.98 Online resources: Table of Contents | Reviews Summary: 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.
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517.98 GEL-G (Browse shelf(Opens below)) Available 25181

Includes bibliographical references and index.

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.

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

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