Generalized Functions, Volume 4

N. Ya. Vilenkin, I.M. Gel'fand

ISBN 10: 1470426625 / ISBN 13: 9781470426620

Editore: American Mathematical Society, US, 2016

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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 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.

Codice articolo LU-9781470426620

Dati bibliografici

• Titolo: Generalized Functions, Volume 4
• Editore: American Mathematical Society, US
• Data di pubblicazione: 2016
• Lingua: Inglese
• ISBN 10: 1470426625
• ISBN 13: 9781470426620
• Rilegatura: Hardback
• Condizione: New
• Peso: 857 grammi

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Riassunto

In this six-volume series on functional analysis, Gel'Fand and Vilkenkin depart from direct sequence for this fourth volume, assuming only the knowledge from the first two chapters of Volume One, and briefly repeating some matters that were discussed in previous chapters. They focus on recent developments in the theory of linear topological spaces, and the construction of harmonic analysis in n-dimensional Euclidean and infinite-dimensional spaces. Their topics are the kernel theorem, nuclear spaces, and rigged Hilbert space; positive and positive-definite generalized functions; generalized random processes; and measures in linear topological spaces. Annotation ©2016 Ringgold, Inc., Portland, OR (protoview.com)