Generalized Functions Volume 2

Author : I. M. Gel'fand,G. E. Shilov
Publisher : American Mathematical Soc.
Page : 261 pages
File Size : 51,6 Mb
Release : 2016-03-30
Category : Theory of distributions (Functional analysis)
ISBN : 9781470426590

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Generalized Functions, Volume 2 by I. M. Gel'fand,G. E. Shilov Pdf

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.

Author : I. M. Gel′fand,G. E. Shilov
Publisher : American Mathematical Soc.
Page : 423 pages
File Size : 50,9 Mb
Release : 2016-04-19
Category : Theory of distributions (Functional analysis)
ISBN : 9781470426583

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Generalized Functions, Volume 1 by I. M. Gel′fand,G. E. Shilov Pdf

he 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 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.

Author : I. M. Gel′fand
Publisher : Unknown
Page : 274 pages
File Size : 51,6 Mb
Release : 1964
Category : Theory of distributions (Functional analysis)
ISBN : 1470431238

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Generalized Functions, Volume 2 by I. M. Gel′fand Pdf

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. Gel2 and 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.

Author : I. M. Gel'Fand,G. E. Shilov
Publisher : Academic Press
Page : 272 pages
File Size : 48,5 Mb
Release : 2013-09-03
Category : Mathematics
ISBN : 9781483262307

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Spaces of Fundamental and Generalized Functions by I. M. Gel'Fand,G. E. Shilov Pdf

Spaces of Fundamental and Generalized Functions, Volume 2, analyzes the general theory of linear topological spaces. The basis of the theory of generalized functions is the theory of the so-called countably normed spaces (with compatible norms), their unions (inductive limits), and also of the spaces conjugate to the countably normed ones or their unions. This set of spaces is sufficiently broad on the one hand, and sufficiently convenient for the analyst on the other. The book opens with a chapter that discusses the theory of these spaces. This is followed by separate chapters on fundamental and generalized functions, Fourier transformations of fundamental and generalized functions, and spaces of type S.

Author : I. M. Gel′fand,M. I. Graev,N. Ya. Vilenkin
Publisher : American Mathematical Soc.
Page : 449 pages
File Size : 46,7 Mb
Release : 2016-04-19
Category : Theory of distributions
ISBN : 9781470426637

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Generalized Functions, Volume 5 by I. M. Gel′fand,M. I. Graev,N. Ya. Vilenkin Pdf

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.

Author : I. M. Gel'fand,G. E. Shilov
Publisher : American Mathematical Soc.
Page : 222 pages
File Size : 54,6 Mb
Release : 2016-03-30
Category : Theory of distributions (Functional analysis)
ISBN : 9781470426613

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Generalized Functions, Volume 3 by I. M. Gel'fand,G. E. Shilov Pdf

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. In Volume 3, applications of generalized functions to the Cauchy problem for systems of partial differential equations with constant coefficients are considered. The book includes the study of uniqueness classes of solutions of the Cauchy problem and the study of classes of functions where the Cauchy problem is well posed. The last chapter of this volume presents results related to spectral decomposition of differential operators related to generalized eigenfunctions.

Author : V.S. Vladimirov,Yu.N. Drozzinov,O.I. Zavialov
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 41,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400928312

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Tauberian Theorems for Generalized Functions by V.S. Vladimirov,Yu.N. Drozzinov,O.I. Zavialov Pdf

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. The Scandal of Father G. K. Chesterton. 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Author : V. S. Vladimirov
Publisher : CRC Press
Page : 332 pages
File Size : 47,6 Mb
Release : 2002-08-15
Category : Mathematics
ISBN : 0415273560

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Methods of the Theory of Generalized Functions by V. S. Vladimirov Pdf

This volume presents the general theory of generalized functions, including the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson integral transforms and operational calculus, with the traditional material augmented by the theory of Fourier series, abelian theorems, and boundary values of helomorphic functions for one and several variables. The author addresses several facets in depth, including convolution theory, convolution algebras and convolution equations in them, homogenous generalized functions, and multiplication of generalized functions. This book will meet the needs of researchers, engineers, and students of applied mathematics, control theory, and the engineering sciences.

Author : Izrail Moiseevitch Gelfand,Georgij Evgenʹevič Šilov,Naum Âkovlevič Vilenkin,Mark Iosifovich Graev
Publisher : Unknown
Page : 1738 pages
File Size : 43,8 Mb
Release : 1964
Category : Electronic
ISBN : LCCN:63016960

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Generalized Functions by Izrail Moiseevitch Gelfand,Georgij Evgenʹevič Šilov,Naum Âkovlevič Vilenkin,Mark Iosifovich Graev Pdf

Author : Michael Grosser
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 50,9 Mb
Release : 2001-11-30
Category : Mathematics
ISBN : 1402001452

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Geometric Theory of Generalized Functions with Applications to General Relativity by Michael Grosser Pdf

This work provides the first comprehensive introduction to the nonlinear theory of generalized functions (in the sense of Colombeau's construction) on differentiable manifolds. Particular emphasis is laid on a diffeomorphism invariant geometric approach to embedding the space of Schwartz distributions into algebras of generalized functions. The foundations of a `nonlinear distributional geometry' are developed, supplying a solid base for an increasing number of applications of algebras of generalized functions to questions of a primarily geometric mature, in particular in mathematical physics. Applications of the resulting theory to symmetry group analysis of differential equations and the theory of general relativity are presented in separate chapters. These features distinguish the present volume from earlier introductory texts and monographs on the subject. Audience: The book will be of interest to graduate students as well as to researchers in functional analysis, partial differential equations, differential geometry, and mathematical physics.

Author : I. M. Gel′fand,N. Ya. Vilenkin
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 49,6 Mb
Release : 2016-04-19
Category : Theory of distributions (Functional analysis)
ISBN : 9781470426620

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Generalized Functions, Volume 4 by I. M. Gel′fand,N. Ya. Vilenkin Pdf

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 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⊂H⊂E′, where H is a Hilbert space, E′ is dual to E, and inclusions E⊂H and H⊂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.

Author : I. M. Gel'fand,N. Ya. Vilenkin
Publisher : Academic Press
Page : 399 pages
File Size : 47,7 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483262246

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Generalized Functions by I. M. Gel'fand,N. Ya. Vilenkin Pdf

Generalized Functions, Volume 4: Applications of Harmonic Analysis is devoted to two general topics—developments in the theory of linear topological spaces and construction of harmonic analysis in n-dimensional Euclidean and infinite-dimensional spaces. This volume specifically discusses the bilinear functionals on countably normed spaces, Hilbert-Schmidt operators, and spectral analysis of operators in rigged Hilbert spaces. The general form of positive generalized functions on the space S, continuous positive-definite functions, and conditionally positive generalized functions are also deliberated. This publication likewise considers the mean of a generalized random process, multidimensional generalized random fields, simplest properties of cylinder sets, and definition of Gaussian measures. This book is beneficial to students, specialists, and researchers aiming to acquire knowledge of functional analysis.

Author : Michael Oberguggenberger,Joachim Toft,Jasson Vindas,Patrik Wahlberg
Publisher : Birkhäuser
Page : 276 pages
File Size : 48,9 Mb
Release : 2017-05-06
Category : Mathematics
ISBN : 9783319519111

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Generalized Functions and Fourier Analysis by Michael Oberguggenberger,Joachim Toft,Jasson Vindas,Patrik Wahlberg Pdf

This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

Author : Dorian Goldfeld,Joseph Hundley
Publisher : Cambridge University Press
Page : 209 pages
File Size : 51,5 Mb
Release : 2011-04-21
Category : Mathematics
ISBN : 9781139503082

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Automorphic Representations and L-Functions for the General Linear Group: Volume 2 by Dorian Goldfeld,Joseph Hundley Pdf

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

Author : Izrail Moiseevitch Gelfand,Georgij Evgenʹevič Šilov,Naum Âkovlevič Vilenkin,Mark Iosifovich Graev
Publisher : Unknown
Page : 1738 pages
File Size : 46,9 Mb
Release : 1964
Category : Electronic
ISBN : LCCN:63016960

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Generalized Functions by Izrail Moiseevitch Gelfand,Georgij Evgenʹevič Šilov,Naum Âkovlevič Vilenkin,Mark Iosifovich Graev Pdf