Generalized Functions, Volume 1

Author: I. M. Gel′fand,G. E. Shilov

Publisher: American Mathematical Soc.

ISBN: 1470426587

Category: Theory of distributions (Functional analysis)

Page: 423

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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.
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Generalized Functions, Volume 6

Author: I. M. Gel′fand,M. I. Graev,I. I. Pyatetskii-Shapiro

Publisher: American Mathematical Soc.

ISBN: 1470426641

Category: Automorphic functions

Page: 426

View: 8452

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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. 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 theme of Volume 6 is the study of representations of the general linear group of order two over various fields and rings of number-theoretic nature, most importantly over local fields (p-adic fields and fields of power series over finite fields) and over the ring of adeles. Representation theory of the latter group naturally leads to the study of automorphic functions and related number-theoretic problems. The book contains a wealth of information about discrete subgroups and automorphic representations, and can be used both as a very good introduction to the subject and as a valuable reference.
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Ultrametric Pseudodifferential Equations and Applications

Author: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

Publisher: Cambridge University Press

ISBN: 1108102905

Category: Mathematics

Page: N.A

View: 722

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Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrametric pseudodifferential equations, including their emerging applications and currently unsolved problems.
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Generalized Functions, Volume 2

Spaces of Fundamental and Generalized Functions

Author: I. M. Gel'fand,G. E. Shilov

Publisher: American Mathematical Soc.

ISBN: 1470426595

Category: Theory of distributions (Functional analysis)

Page: 261

View: 3578

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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. 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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Generalized Functions, Volume 5

Author: I. M. Gel′fand,M. I. Graev,N. Ya. Vilenkin

Publisher: American Mathematical Soc.

ISBN: 1470426633

Category: Theory of distributions (Functional analysis)

Page: 449

View: 9806

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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. 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.
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Automorphic Functions

Author: Lester R. Ford

Publisher: American Mathematical Soc.

ISBN: 9780821837412

Category: Mathematics

Page: 333

View: 2619

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Lester Ford's book was the first treatise in English on automorphic functions. At the time of its publication (1929), it was welcomed for its elegant treatment of groups of linear transformations and for the remarkably clear and explicit exposition throughout the book. Ford's extraordinary talent for writing has been memorialized in the prestigious award that bears his name. The book, in the meantime, has become a recognized classic. Ford's approach is primarily through analytic function theory. The first part of the book covers groups of linear transformations, especially Fuchsian groups, fundamental domains, and functions that are invariant under the groups, including the classical elliptic modular functions and Poincare theta series. The second part of the book covers conformal mappings, uniformization, and connections between automorphic functions and differential equations with regular singular points, such as the hypergeometric equation.
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Electrical Papers

Author: Oliver Heaviside

Publisher: American Mathematical Soc.

ISBN: 9780821834640

Category: Mathematics

Page: 560

View: 927

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Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.
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