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Generalized Functions With New Understanding And Forms
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Book Synopsis Generalized Functions, Volume 2 by : I. M. Gel'fand
Download or read book Generalized Functions, Volume 2 written by I. M. Gel'fand and published by American Mathematical Soc.. This book was released on 2016-03-30 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Generalized Functions Theory and Technique by : Ram P. Kanwal
Download or read book Generalized Functions Theory and Technique written by Ram P. Kanwal and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition, there has been tremendous growth in the subject and I have attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12 of this edition by a judicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application of the probability theory has been revised, and I am thankful to Professor Z.L. Crvenkovic for her help. The new material included in this chapter pertains to the modern topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.
Book Synopsis Methods of the Theory of Generalized Functions by : V. S. Vladimirov
Download or read book Methods of the Theory of Generalized Functions written by V. S. Vladimirov and published by CRC Press. This book was released on 2002-08-15 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Distribution Theory and Transform Analysis by : A.H. Zemanian
Download or read book Distribution Theory and Transform Analysis written by A.H. Zemanian and published by Courier Corporation. This book was released on 2011-11-30 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Distribution theory, a relatively recent mathematical approach to classical Fourier analysis, not only opened up new areas of research but also helped promote the development of such mathematical disciplines as ordinary and partial differential equations, operational calculus, transformation theory, and functional analysis. This text was one of the first to give a clear explanation of distribution theory; it combines the theory effectively with extensive practical applications to science and engineering problems. Based on a graduate course given at the State University of New York at Stony Brook, this book has two objectives: to provide a comparatively elementary introduction to distribution theory and to describe the generalized Fourier and Laplace transformations and their applications to integrodifferential equations, difference equations, and passive systems. After an introductory chapter defining distributions and the operations that apply to them, Chapter 2 considers the calculus of distributions, especially limits, differentiation, integrations, and the interchange of limiting processes. Some deeper properties of distributions, such as their local character as derivatives of continuous functions, are given in Chapter 3. Chapter 4 introduces the distributions of slow growth, which arise naturally in the generalization of the Fourier transformation. Chapters 5 and 6 cover the convolution process and its use in representing differential and difference equations. The distributional Fourier and Laplace transformations are developed in Chapters 7 and 8, and the latter transformation is applied in Chapter 9 to obtain an operational calculus for the solution of differential and difference equations of the initial-condition type. Some of the previous theory is applied in Chapter 10 to a discussion of the fundamental properties of certain physical systems, while Chapter 11 ends the book with a consideration of periodic distributions. Suitable for a graduate course for engineering and science students or for a senior-level undergraduate course for mathematics majors, this book presumes a knowledge of advanced calculus and the standard theorems on the interchange of limit processes. A broad spectrum of problems has been included to satisfy the diverse needs of various types of students.
Book Synopsis Handbook of Function and Generalized Function Transformations by : Ahmed I. Zayed
Download or read book Handbook of Function and Generalized Function Transformations written by Ahmed I. Zayed and published by CRC Press. This book was released on 1996-05-15 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: Function transformations, which include linear integral transformations, are some of the most important mathematical tools for solving problems in all areas of engineering and the physical sciences. They allow one to quickly solve a problem by breaking it down into a series of smaller, more manageable problems. The author has compiled the most important and widely used of these function transforms in applied mathematics and electrical engineering. In addition to classical transforms, newer transforms such as wavelets, Zak, and Radon are included. The book is neither a table of transforms nor a textbook, but it is a source book that provides quick and easy access to the most important properties and formulas of function and generalized function transformations. It is organized for convenient reference, with chapters broken down into the following sections:
Book Synopsis Generalized Calculus with Applications to Matter and Forces by : Luis Manuel Braga de Costa Campos
Download or read book Generalized Calculus with Applications to Matter and Forces written by Luis Manuel Braga de Costa Campos and published by CRC Press. This book was released on 2014-04-18 with total page 888 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as: Gauss and Stokes’ theorems in the differential geometry, tensor calculus, and theory of potential fields Self-adjoint and non-self-adjoint problems for linear differential equations and nonlinear problems with large deformations Multipolar expansions and Green’s functions for elastic strings and bars, potential and rotational flow, electro- and magnetostatics, and more This third volume in the series Mathematics and Physics for Science and Technology is designed to complete the theory of functions and its application to potential fields, relating generalized functions to broader follow-on topics like differential equations. Featuring step-by-step examples with interpretations of results and discussions of assumptions and their consequences, Generalized Calculus with Applications to Matter and Forces enables readers to construct mathematical–physical models suited to new observations or novel engineering devices.
Book Synopsis Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics by : F. Farassat
Download or read book Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics written by F. Farassat and published by . This book was released on 1994 with total page 60 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Generalized Functions by : Ram P. Kanwal
Download or read book Generalized Functions written by Ram P. Kanwal and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides a more cohesive and sharply focused treatment of fundamental concepts and theoretical background material, with particular attention given to better delineating connections to varying applications Exposition driven by additional examples and exercises
Book Synopsis Generalized Functions and Their Applications by : R.S. Pathak
Download or read book Generalized Functions and Their Applications written by R.S. Pathak and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: The International Symposium on Generalized Functions and Their Applications was organized by the Department of Mathematics, Banaras Hindu University, and held December 23-26, 1991, on the occasion of the Platinum Jubilee Celebration of the university. More than a hundred mathematicians from ten countries participated in the deliberations of the symposium. Thirty lectures were delivered on a variety of topics within the area. The contributions to the proceedings of the symposium are, with a few exceptions, expanded versions of the lectures delivered by the invited speakers. The survey papers by Komatsu and Hoskins and Sousa Pinto provide an up-to-date account of the theory of hyperfunctions, ultradistributions and microfunctions, and the nonstandard theory of new generalized functions, respectively; those by Stankovic and Kanwal deal with structures and asymptotics. Choquet-Bruhat's work studies generalized functions on manifold and gives applications to shocks and discrete models. The other contributions relate to contemporary problems and achievements in theory and applications, especially in the theory of partial differential equations, differential geometry, mechanics, mathematical physics, and systems science. The proceedings give a very clear impression of the present state of the art in this field and contain many challenges, ideas, and open problems. The volume is very helpful for a broad spectrum of readers: graduate students to mathematical researchers.
Book Synopsis Geometric Theory of Generalized Functions with Applications to General Relativity by : M. Grosser
Download or read book Geometric Theory of Generalized Functions with Applications to General Relativity written by M. Grosser and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past few years a certain shift of focus within the theory of algebras of generalized functions (in the sense of J. F. Colombeau) has taken place. Originating in infinite dimensional analysis and initially applied mainly to problems in nonlinear partial differential equations involving singularities, the theory has undergone a change both in in ternal structure and scope of applicability, due to a growing number of applications to questions of a more geometric nature. The present book is intended to provide an in-depth presentation of these develop ments comprising its structural aspects within the theory of generalized functions as well as a (selective but, as we hope, representative) set of applications. This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here. First, despite the fact that by now several excellent mono graphs on Colombeau algebras are available, we have decided to give a self-contained introduction to the field in Chapter 1. Our motivation for this decision derives from two main features of our approach. On the one hand, in contrast to other treatments of the subject we base our intro duction to the field on the so-called special variant of the algebras, which makes many of the fundamental ideas of the field particularly transpar ent and at the same time facilitates and motivates the introduction of the more involved concepts treated later in the chapter.
Book Synopsis Generalized Functions by : I. M. Gel'fand
Download or read book Generalized Functions written by I. M. Gel'fand and published by Academic Press. This book was released on 2014-05-12 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Generalized Functions and Partial Differential Equations by : Avner Friedman
Download or read book Generalized Functions and Partial Differential Equations written by Avner Friedman and published by Courier Corporation. This book was released on 2005-12-10 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research. Geared toward upper-level undergraduates and graduate students, the text assumes a sound knowledge of both real and complex variables. Familiarity with the basic theory of functional analysis, especially normed spaces, is helpful but not necessary. An introductory chapter features helpful background on topological spaces. Applications to partial differential equations include a treatment of the Cauchy problem, the Goursat problem, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. Supplementary materials include end-of-chapter problems, bibliographical remarks, and a bibliography.
Book Synopsis Asymptotic Behavior of Generalized Functions by : Stevan Pilipovi?
Download or read book Asymptotic Behavior of Generalized Functions written by Stevan Pilipovi? and published by World Scientific. This book was released on 2012 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: The asymptotic analysis has obtained new impulses with the general development of various branches of mathematical analysis and their applications. In this book, such impulses originate from the use of slowly varying functions and the asymptotic behavior of generalized functions. The most developed approaches related to generalized functions are those of Vladimirov, Drozhinov and Zavyalov, and that of Kanwal and Estrada. The first approach is followed by the authors of this book and extended in the direction of the S-asymptotics. The second approach ? of Estrada, Kanwal and Vindas ? is related to moment asymptotic expansions of generalized functions and the Ces'aro behavior. The main features of this book are the uses of strong methods of functional analysis and applications to the analysis of asymptotic behavior of solutions to partial differential equations, Abelian and Tauberian type theorems for integral transforms as well as for the summability of Fourier series and integrals. The book can be used by applied mathematicians, physicists, engineers and others who use classical asymptotic methods and wish to consider non-classical objects (generalized functions) and their asymptotics now in a more advanced setting.
Book Synopsis Generalized Functions, Volume 4 by : I. M. Gel′fand
Download or read book Generalized Functions, Volume 4 written by I. M. Gel′fand and published by American Mathematical Soc.. This book was released on 2016-04-19 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Theories of Generalised Functions by : R F Hoskins
Download or read book Theories of Generalised Functions written by R F Hoskins and published by Elsevier. This book was released on 2005-01-01 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explaining and comparing the various standard types of generalised functions which have been developed during the 20th Century, this text also contains accounts of recent non-standard theories of distributions, ultradistributions and Stato-hyperfunctions. The book could readily be used as a main text on generalised functions for mathematical undergraduates in final year analysis courses, as it presupposes little more than a general mathematical background. It also makes a valuable reference text for non-specific applied mathematics students, such as physicists or electrical engineers, needing to gain expertise in the application of generalised functions to physical problems, without any prior acquaintance of the specialised subject matter. An ideal companion book to Delta Functions, also by Professor Hoskins. - Explains and compares the various standard types of generalised functions that have been developed during the 20th Century - Contains accounts of recent non-standard theories of distributions, ultradistributions and Stato-hyperfunctions
Book Synopsis Handbook of Function and Generalized Function Transformations by : Ahmed I. Zayed
Download or read book Handbook of Function and Generalized Function Transformations written by Ahmed I. Zayed and published by CRC Press. This book was released on 2019-08-21 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: Function transformations, which include linear integral transformations, are some of the most important mathematical tools for solving problems in all areas of engineering and the physical sciences. They allow one to quickly solve a problem by breaking it down into a series of smaller, more manageable problems. The author has compiled the most important and widely used of these function transforms in applied mathematics and electrical engineering. In addition to classical transforms, newer transforms such as wavelets, Zak, and Radon are included. The book is neither a table of transforms nor a textbook, but it is a source book that provides quick and easy access to the most important properties and formulas of function and generalized function transformations.
Book Synopsis Generalized Functions, Volume 1 by : I. M. Gel′fand
Download or read book Generalized Functions, Volume 1 written by I. M. Gel′fand and published by American Mathematical Soc.. This book was released on 2016-04-19 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
