An Extension Of The Laplace Transform To Schwartz Distributions
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An Extension Of The Laplace Transform To Schwartz Distributions
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Author : Douglas B. Price
language : en
Publisher:
Release Date : 1974
An Extension Of The Laplace Transform To Schwartz Distributions written by Douglas B. Price and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Laplace transformation categories.
A new characterization of the Laplace transform is developed which extends the transform to the Schwartz distributions. The class of distributions includes, in addition to all ordinary functions, the impulse functions and other singular functions which occur as solutions to ordinary and partial differential equations. The standard theorems on analyticity, uniqueness, and invertibility of the transform are proved by using the new characterization as the definition of the Laplace transform. The new definition uses sequences of linear transformations on the space of distributions in a manner suggested by a paper of E. Gesztelyi which extended the Laplace transform to another class of generalized functions, the Mikusinski operators. It is shown that the new sequential definition of the transform is equivalent to Schwartz' extension of the ordinary Laplace transform to distributions but, in contrast to Schwartz' definition, does not use the distributional Fourier transform. Several theorems concerning the particular linear transformations used to define the Laplace transforms are proved. All the results proved in one dimension are extended to the n-dimensional case, but proofs are presented only for those situations that require methods different from their one-dimensional analogs.
Nasa Technical Report
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Author : United States. National Aeronautics and Space Administration
language : en
Publisher:
Release Date : 1973
Nasa Technical Report written by United States. National Aeronautics and Space Administration and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973 with Aerodynamics categories.
Applied Mechanics Reviews
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Author :
language : en
Publisher:
Release Date : 1974
Applied Mechanics Reviews written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Mechanics, Applied categories.
Scientific And Technical Aerospace Reports
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Author :
language : en
Publisher:
Release Date : 1992
Scientific And Technical Aerospace Reports written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Aeronautics categories.
Distributional Integral Transforms
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Author : P.K. Banerjee
language : en
Publisher: Scientific Publishers
Release Date : 2005-09-01
Distributional Integral Transforms written by P.K. Banerjee and has been published by Scientific Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-09-01 with Mathematics categories.
The present Learned Research Work is an exhaustive survey and researches carried out by the authors, which led to the theories of distributions, generalized functions and transforms involving them, which includes interesting results and the fundamental concepts of the youngest generalization of Schwartz theory of distributions, the Boehmians. The tempered distribution and utilizations have been described, which provide suitable platforms for the generalizations of Fourier transforms, Stieltjes and Mellin transforms. To overcome the Fourier series this work includes wavelet transform, for which meticulous extensive study of the existing literature has been produced including recent researches carried out by the authors. This compilation, in the form of the present book, is believed to be of help to researchers in the field of distribution and transform analysis and, may even be treated as the reference book to post graduate students.
Fourier And Laplace Transforms
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Author :
language : en
Publisher: Cambridge University Press
Release Date : 2003-08-07
Fourier And Laplace Transforms written by and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-08-07 with Mathematics categories.
This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science.
Monthly Catalog Of United States Government Publications
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Author :
language : en
Publisher:
Release Date : 1974
Monthly Catalog Of United States Government Publications written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Government publications categories.
Introduction To The Theory And Application Of The Laplace Transformation
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Author : G. Doetsch
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Introduction To The Theory And Application Of The Laplace Transformation written by G. Doetsch and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. Chiefly, they treat problems which, in mathematical language, are governed by ordi nary and partial differential equations, in various physically dressed forms. The theoretical foundations of the Laplace transformation are presented usually only in a simplified manner, presuming special properties with respect to the transformed func tions, which allow easy proofs. By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists a,nd engineers in their daily routine work, but in complete generality and with detailed, exact proofs. The applications to other mathematical domains and to technical prob lems are inserted, when the theory is adequately· developed to present the tools necessary for their treatment. Since the book proceeds, not in a rigorously systematic manner, but rather from easier to more difficult topics, it is suited to be read from the beginning as a textbook, when one wishes to familiarize oneself for the first time with the Laplace transforma tion. For those who are interested only in particular details, all results are specified in "Theorems" with explicitly formulated assumptions and assertions. Chapters 1-14 treat the question of convergence and the mapping properties of the Laplace transformation. The interpretation of the transformation as the mapping of one function space to another (original and image functions) constitutes the dom inating idea of all subsequent considerations.
Distribution Theory And Transform Analysis
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Author : A.H. Zemanian
language : en
Publisher: Courier Corporation
Release Date : 2011-11-30
Distribution Theory And Transform Analysis written by A.H. Zemanian and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-11-30 with Mathematics categories.
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.
Mathematical Tools For Physicists
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Author : Michael Grinfeld
language : en
Publisher: John Wiley & Sons
Release Date : 2015-01-12
Mathematical Tools For Physicists written by Michael Grinfeld and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-12 with Science categories.
The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.