Generalised Functions

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Generalized Functions Theory and Technique

Author : Ram P. Kanwal
Publisher : Springer Science & Business Media
ISBN 13 : 1468400355
Total Pages : 474 pages
Book Rating : 4.4/5 (684 download)

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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.

Geometric Theory of Generalized Functions with Applications to General Relativity

Author : M. Grosser
Publisher : Springer Science & Business Media
ISBN 13 : 9401598452
Total Pages : 517 pages
Book Rating : 4.4/5 (15 download)

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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.

Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum Mechanics

Author : W.-H. Steeb
Publisher : Springer Science & Business Media
ISBN 13 : 9401153329
Total Pages : 247 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum Mechanics by : W.-H. Steeb

Download or read book Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum Mechanics written by W.-H. Steeb and published by Springer Science & Business Media. This book was released on 2013-03-07 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a comprehensive introduction to modern quantum mechanics, emphasising the underlying Hilbert space theory and generalised function theory. All the major modern techniques and approaches used in quantum mechanics are introduced, such as Berry phase, coherent and squeezed states, quantum computing, solitons and quantum mechanics. Audience: The book is suitable for graduate students in physics and mathematics.

Methods of the Theory of Generalized Functions

Author : V. S. Vladimirov
Publisher : CRC Press
ISBN 13 : 9780415273565
Total Pages : 332 pages
Book Rating : 4.2/5 (735 download)

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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.

An Introduction to Fourier Analysis and Generalised Functions

Author : M. J. Lighthill
Publisher :
ISBN 13 :
Total Pages : 96 pages
Book Rating : 4.:/5 (318 download)

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Book Synopsis An Introduction to Fourier Analysis and Generalised Functions by : M. J. Lighthill

Download or read book An Introduction to Fourier Analysis and Generalised Functions written by M. J. Lighthill and published by . This book was released on 1958 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Clearly and attractively written, but without any deviation from rigorous standards of mathematical proof...." Science Progress

The Theory of Generalised Functions

Author : D. S. Jones
Publisher : Cambridge University Press
ISBN 13 : 9780521100045
Total Pages : 0 pages
Book Rating : 4.1/5 ( download)

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Book Synopsis The Theory of Generalised Functions by : D. S. Jones

Download or read book The Theory of Generalised Functions written by D. S. Jones and published by Cambridge University Press. This book was released on 2009-01-18 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting from an elementary level Professor Jones discusses generalised functions and their applications. He aims to supply the simplest introduction for those who wish to learn to use generalised functions and there is liberal provision of exercises with which to gain experience. The study of more advanced topics such as partial differential equations, Laplace transforms and ultra-distributions should also make it a valuable source for researchers. The demands placed upon the reader's analytical background are the minimum required to approach this topic. Therefore, by selecting chapters it is possible to construct a short introductory course for students, a final-year option for honours undergraduates or a comprehensive postgraduate course.

Theories of Generalised Functions

Author : R F Hoskins
Publisher : Elsevier
ISBN 13 : 0857099485
Total Pages : 307 pages
Book Rating : 4.8/5 (57 download)

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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

An Introduction to Fourier Analysis and Generalised Functions

Author : Sir M. J. Lighthill
Publisher : Cambridge University Press
ISBN 13 : 9780521091282
Total Pages : 112 pages
Book Rating : 4.0/5 (912 download)

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Book Synopsis An Introduction to Fourier Analysis and Generalised Functions by : Sir M. J. Lighthill

Download or read book An Introduction to Fourier Analysis and Generalised Functions written by Sir M. J. Lighthill and published by Cambridge University Press. This book was released on 1958 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Clearly and attractively written, but without any deviation from rigorous standards of mathematical proof...." Science Progress

Eigenvalues, Embeddings and Generalised Trigonometric Functions

Author : Jan Lang
Publisher : Springer Science & Business Media
ISBN 13 : 3642182674
Total Pages : 232 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Eigenvalues, Embeddings and Generalised Trigonometric Functions by : Jan Lang

Download or read book Eigenvalues, Embeddings and Generalised Trigonometric Functions written by Jan Lang and published by Springer Science & Business Media. This book was released on 2011-03-23 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone decreasing sequence of non-negative numbers with a view to the classification of operators according to the way in which these numbers approach a limit: approximation numbers provide an especially important example of such numbers. The asymptotic behavior of the s-numbers of Hardy operators acting between Lebesgue spaces is determined here in a wide variety of cases. The proof methods involve the geometry of Banach spaces and generalized trigonometric functions; there are connections with the theory of the p-Laplacian.

Generalized Functions and Fourier Analysis

Author : Michael Oberguggenberger
Publisher : Birkhäuser
ISBN 13 : 3319519115
Total Pages : 280 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Generalized Functions and Fourier Analysis by : Michael Oberguggenberger

Download or read book Generalized Functions and Fourier Analysis written by Michael Oberguggenberger and published by Birkhäuser. This book was released on 2017-05-06 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Spaces of Fundamental and Generalized Functions

Author : I. M. Gel'Fand
Publisher : Academic Press
ISBN 13 : 1483262308
Total Pages : 272 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Spaces of Fundamental and Generalized Functions by : I. M. Gel'Fand

Download or read book Spaces of Fundamental and Generalized Functions written by I. M. Gel'Fand and published by Academic Press. This book was released on 2013-09-03 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Delta Functions

Author : R F Hoskins
Publisher : Horwood Publishing
ISBN 13 : 9781904275398
Total Pages : 284 pages
Book Rating : 4.2/5 (753 download)

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Book Synopsis Delta Functions by : R F Hoskins

Download or read book Delta Functions written by R F Hoskins and published by Horwood Publishing. This book was released on 2009-04-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: Delta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions for the first time. In particular, the treatment of the Laplace transform has been revised with this in mind. The chapter on Schwartz distributions has been considerably extended and the book is supplemented by a fuller review of Nonstandard Analysis and a survey of alternative infinitesimal treatments of generalised functions. Dealing with a difficult subject in a simple and straightforward way, the text is readily accessible to a broad audience of scientists, mathematicians and engineers. It can be used as a working manual in its own right, and serves as a preparation for the study of more advanced treatises. Little more than a standard background in calculus is assumed, and attention is focused on techniques, with a liberal selection of worked examples and exercises. Second edition has been updated, restructured and modernised to answer specific difficulties typically found by students encountering delta functions for the first time Attention is focused on techniques, with a liberal selection of worked examples and exercises Readily accessible to a broad audience of scientists, mathematicians and engineers and can be used as a working manual in its own right

Distribution Theory and Transform Analysis

Author : A.H. Zemanian
Publisher : Courier Corporation
ISBN 13 : 0486151948
Total Pages : 404 pages
Book Rating : 4.4/5 (861 download)

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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.

Generalized Hypergeometric Functions

Author : Lucy Joan Slater
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (493 download)

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Book Synopsis Generalized Hypergeometric Functions by : Lucy Joan Slater

Download or read book Generalized Hypergeometric Functions written by Lucy Joan Slater and published by . This book was released on 1993 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Generalized Associated Legendre Functions and Their Applications

Author : Nina Opanasivna Virchenko
Publisher : World Scientific
ISBN 13 : 9810243537
Total Pages : 217 pages
Book Rating : 4.8/5 (12 download)

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Book Synopsis Generalized Associated Legendre Functions and Their Applications by : Nina Opanasivna Virchenko

Download or read book Generalized Associated Legendre Functions and Their Applications written by Nina Opanasivna Virchenko and published by World Scientific. This book was released on 2001 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: The various types of special functions have become essential tools for scientists and engineers. One of the important classes of special functions is of the hypergeometric type. It includes all classical hypergeometric functions such as the well-known Gaussian hypergeometric functions, the Bessel, Macdonald, Legendre, Whittaker, Kummer, Tricomi and Wright functions, the generalized hypergeometric functions ?Fq, Meijer's G-function, Fox's H-function, etc.Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed form, to examine these solutions, and to investigate the relationships between different classes of the special functions.This book deals with the theory and applications of generalized associated Legendre functions of the first and the second kind, Pm, n?(z) and Qm, n?(z), which are important representatives of the hypergeometric functions. They occur as generalizations of classical Legendre functions of the first and the second kind respectively. The authors use various methods of contour integration to obtain important properties of the generalized associated Legnedre functions as their series representations, asymptotic formulas in a neighborhood of singular points, zero properties, connection with Jacobi functions, Bessel functions, elliptic integrals and incomplete beta functions.The book also presents the theory of factorization and composition structure of integral operators associated with the generalized associated Legendre function, the fractional integro-differential properties of the functions Pm, n?(z) and Qm, n?(z), the classes of dual and triple integral equations associated with the function Pm, n-1/2+i?(chà) etc.

Nonlinear Theory of Generalized Functions

Author : Michael Oberguggenberger
Publisher : Routledge
ISBN 13 : 1351428039
Total Pages : 400 pages
Book Rating : 4.3/5 (514 download)

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Book Synopsis Nonlinear Theory of Generalized Functions by : Michael Oberguggenberger

Download or read book Nonlinear Theory of Generalized Functions written by Michael Oberguggenberger and published by Routledge. This book was released on 2022-02-28 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis. The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.

Generalized Functions and Their Applications

Author : R.S. Pathak
Publisher : Springer Science & Business Media
ISBN 13 : 1489915915
Total Pages : 298 pages
Book Rating : 4.4/5 (899 download)

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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.