A Guide To Distribution Theory And Fourier Transforms
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A Guide to Distribution Theory and Fourier Transforms by Robert S. Strichartz Pdf
This important book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudodifferential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book.A Guide to Distribution Theory and Fourier Transforms can serve as a textbook for parts of a course on Applied Analysis or Methods of Mathematical Physics, and in fact it is used that way at Cornell.
Distribution Theory and Transform Analysis by A.H. Zemanian Pdf
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.
The theory of distributions has numerous applications and is extensively used in mathematics, physics and engineering. There is however relatively little elementary expository literature on distribution theory. This book is intended as an introduction. Starting with the elementary theory of distributions, it proceeds to convolution products of distributions, Fourier and Laplace transforms, tempered distributions, summable distributions and applications. The theory is illustrated by several examples, mostly beginning with the case of the real line and then followed by examples in higher dimensions. This is a justified and practical approach, it helps the reader to become familiar with the subject. A moderate number of exercises are added. It is suitable for a one-semester course at the advanced undergraduate or beginning graduate level or for self-study.
Distributions by J.J. Duistermaat,Johan A.C. Kolk Pdf
This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.
A Course in Distribution Theory and Applications by R. S. Pathak Pdf
The book covers important topics: basic properties of distributions, convolution, Fourier transforms, Sobolev spaces, weak solutions, distributions on locally convex spaces and on differentiable manifolds. It is a largely self-contained text.".
Distribution Theory and Applications by Abdellah El Kinani,Mohamed Oudadess Pdf
The general frame for the resolution of PDEs is the theory of kernels ù the first elements of which are sufficient to show the practicality of distribution theory in applications. --
Distributions, Fourier Transforms And Some Of Their Applications To Physics by Schucker Thomas Pdf
In this book, distributions are introduced via sequences of functions. This approach due to Temple has two virtues:The Fourier transform is defined for functions and generalized to distributions, while the Green function is defined as the outstanding application of distributions. Using Fourier transforms, the Green functions of the important linear differential equations in physics are computed. Linear algebra is reviewed with emphasis on Hilbert spaces. The author explains how linear differential operators and Fourier transforms naturally fit into this frame, a point of view that leads straight to generalized fourier transforms and systems of special functions like spherical harmonics, Hermite, Laguerre, and Bessel functions.
This new edition of an indispensable text provides a clear treatment of Fourier Series, Fourier Transforms, and FFTs. The unique software, included with the book and newly updated for this edition, allows the reader to generate, firsthand, images of all aspects of Fourier analysis described in the text. Topics covered include :
Distributions and Their Applications in Physics by F. Constantinescu Pdf
Distributions and Their Applications in Physics is the introduction of the Theory of Distributions and their applications in physics. The book contains a discussion of those topics under the Theory of Distributions that are already considered classic, which include local distributions; distributions with compact support; tempered distributions; the distribution theory in relativistic physics; and many others. The book also covers the Normed and Countably-normed Spaces; Test Function Spaces; Distribution Spaces; and the properties and operations involved in distributions. The text is recommended for physicists that wish to be acquainted with distributions and their relevance and applications as part of mathematical and theoretical physics, and for mathematicians who wish to be acquainted with the application of distributions theory for physics.
The Prehistory of the Theory of Distributions by J. Lützen Pdf
I first learned the theory of distributions from Professor Ebbe Thue Poulsen in an undergraduate course at Aarhus University. Both his lectures and the textbook, Topological Vector Spaces, Distributions and Kernels by F. Treves, used in the course, opened my eyes to the beauty and abstract simplicity of the theory. However my incomplete study of many branches of classical analysis left me with the question: Why is the theory of distributions important? In my continued studies this question was gradually answered, but my growing interest in the history of mathematics caused me to alter my question to other questions such as: For what purpose, if any, was the theory of distributions originally created? Who invented distributions and when? I quickly found answers to the last two questions: distributions were invented by S. Sobolev and L. Schwartz around 1936 and 1950, respectively. Knowing this answer, however, only created a new question: Did Sobolev and Schwartz construct distributions from scratch or were there earlier trends and, if so, what were they? It is this question, concerning the pre history of the theory of distributions, which I attempt to answer in this book. Most of my research took place at the History of Science Department of Aarhus University. I wish to thank this department for its financial and intellectual support. I am especially grateful to Lektors Kirsti Andersen from the History of Science Department and Lars Mejlbo from the Mathematics Department, for their kindness, constructive criticism, and encouragement.
A Student's Guide to Fourier Transforms by J. F. James Pdf
Fourier transform theory is of central importance in a vast range of applications in physical science, engineering and applied mathematics. Providing a concise introduction to the theory and practice of Fourier transforms, this book is invaluable to students of physics, electrical and electronic engineering, and computer science. After a brief description of the basic ideas and theorems, the power of the technique is illustrated through applications in optics, spectroscopy, electronics and telecommunications. The rarely discussed but important field of multi-dimensional Fourier theory is covered, including a description of Computer Axial Tomography (CAT scanning). The book concludes by discussing digital methods, with particular attention to the Fast Fourier Transform and its implementation. This new edition has been revised to include new and interesting material, such as convolution with a sinusoid, coherence, the Michelson stellar interferometer and the van Cittert–Zernike theorem, Babinet's principle and dipole arrays.
In a simple but mathematically coherent manner, this text examines the basis of the distribution theories devised by Schwartz and by Mikusinki. Rigorous and concise, it surveys the functional theory of distributions as well as the algebraic theory. Easy generalizations offer applications to a wide variety of problems. 1962 edition.
Distribution Theory by Petre Teodorescu,Wilhelm W. Kecs,Antonela Toma Pdf
In this comprehensive monograph, the authors apply modern mathematical methods to the study of mechanical and physical phenomena or techniques in acoustics, optics, and electrostatics, where classical mathematical tools fail. They present a general method of approaching problems, pointing out different aspects and difficulties that may occur. With respect to the theory of distributions, only the results and the principle theorems are given as well as some mathematical results. The book also systematically deals with a large number of applications to problems of general Newtonian mechanics, as well as to problems pertaining to the mechanics of deformable solids and physics. Special attention is placed upon the introduction of corresponding mathematical models. Addressed to a wide circle of readers who use mathematical methods in their work: applied mathematicians, engineers in various branches, as well as physicists, while also benefiting students in various fields.
