Transform Analysis of Generalized Functions concentrates on finite parts of integrals, generalized functions and distributions. It gives a unified treatment of the distributional setting with transform analysis, i.e. Fourier, Laplace, Stieltjes, Mellin, Hankel and Bessel Series. Included are accounts of applications of the theory of integral transforms in a distributional setting to the solution of problems arising in mathematical physics. Information on distributional solutions of differential, partial differential equations and integral equations is conveniently collected here. The volume will serve as introductory and reference material for those interested in analysis, applications, physics and engineering.
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.
For those who have a background in advanced calculus, elementary topology and functional analysis - from applied mathematicians and engineers to physicists - researchers and graduate students alike - this work provides a comprehensive analysis of the many important integral transforms and renders particular attention to all of the technical aspects of the subject. The author presents the last two decades of research and includes important results from other works.
English translation (from revised and enlarged versions of the Russian editions of 1977 and 1984) of a reference work which makes available to engineers, physicists and applied mathematicians theoretical and tabular material pertaining to certain extensions of standard integral transform techniques. Diverse transforms are touched upon, but the emphasis (particularly in the tables) is on generalized Fourier and Laplace transforms. Some multi-dimensional results are presented. Expensive, but nicely produced, and redundant with nothing standard to the reference shelves of mathematical libraries. (NW) Annotation copyrighted by Book News, Inc., Portland, OR
The generalized function is one of the important branches of mathematics which has enormous applications in practical fields. In particular its applications to the theory of distribution and signal processing are very much essential. In this computer age, information science plays a very important role and the Fourier transform is extremely significant in deciphering obscured information to be made understandable. The book contains six chapters and three appendices. Chapter 1 deals with the preliminary remarks of Fourier series from general point of view. Chapter 2 is concerned with the generalized functions and their Fourier transforms. Chapter 3 contains the Fourier transforms of particular generalized functions. Chapter 4 deals with the asymptotic estimation of Fourier transforms. Chapter 5 is devoted to the study of Fourier series as a series of generalized functions. Chapter 6 deals with the fast Fourier transforms.Appendix A contains the extended list of Fourier transform pairs.Appendix B illustrates the properties of impulse function.Appendix C contains an extended list of biographical references
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
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:
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.
For those who have a background in advanced calculus, elementary topology and functional analysis - from applied mathematicians and engineers to physicists - researchers and graduate students alike - this work provides a comprehensive analysis of the many important integral transforms and renders particular attention to all of the technical aspects of the subject. The author presents the last two decades of research and includes important results from other works.
