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This book covers the basic elements of difference equations andthe tools of difference and sum calculus necessary for studying andsolving, primarily, ordinary linear difference equations. It islucidly written and carefully motivated with examples from variousfields of applications. These examples are presented in the firstchapter and then discussed with their detailed solutions in Chapters2-7. A particular feature is the use of the discrete Fouriertransforms for solving difference equations associated with, generallynonhomogeneous, boundary conditions. Emphasis is placed onillustrating this new method by means of applications. The primarygoal of the book is to serve as a primer for a first course in lineardifference equations but, with the addition of more theory andapplications, the book is suitable for more advanced courses.
Audience: In addition to students from mathematics and appliedfields the book will be of value to academic and industrialresearchers who are interested in applications. ... Read more

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This reference/text desribes the basic elements of the integral, finite, and discrete transforms - emphasizing their use for solving boundary and initial value problems as well as facilitating the representations of signals and systems.;Proceeding to the final solution in the same setting of Fourier analysis without interruption, Integral and Discrete Transforms with Applications and Error Analysis: presents the background of the FFT and explains how to choose the appropriate transform for solving a boundary value problem; discusses modelling of the basic partial differential equations, as well as the solutions in terms of the main special functions; considers the Laplace, Fourier, and Hankel transforms and their variations, offering a more logical continuation of the operational method; covers integral, discrete, and finite transforms and trigonometric Fourier and general orthogonal series expansion, providing an application to signal analysis and boundary-value problems; and examines the practical approximation of computing the resulting Fourier series or integral representation of the final solution and treats the errors incurred.;Containing many detailed examples and numerous end-of-chapter exercises of varying difficulty for each section with answers, Integral and Discrete Transforms with Applications and Error Analysis is a thorough reference for analysts; industrial and applied mathematicians; electrical, electronics, and other engineers; and physicists and an informative text for upper-level undergraduate and graduate students in these disciplines. ... Read more

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This volume contains an introduction to the Picard--Lefschetztheory, which controls the ramification and qualitative behaviour ofmany important functions of PDEs and integral geometry, and itsfoundations in singularity theory. Solutions to many problems of these theories are treated. Subjectsinclude the proof of multidimensional analogues of Newton's theorem onthe nonintegrability of ovals; extension of the proofs for thetheorems of Newton, Ivory, Arnold and Givental on potentials ofalgebraic surfaces. Also, it is discovered for which dandn the potentials of degree d hyperbolic surfaces inn are algebraic outside the surfaces; the equivalence oflocal regularity (the so-called sharpness), of fundamental solutionsof hyperbolic PDEs and the topologicalPetrovskii--Atiyah--Bott--Garding condition is proved, and thegeometrical characterization of domains of sharpness close to simplesingularities of wave fronts is considered; a `stratified' version ofthe Picard--Lefschetz formula is proved, and an algorithm enumeratingtopologically distinct Morsifications of real function singularitiesis given. This book will be valuable to those who are interested in integraltransforms, operational calculus, algebraic geometry, PDEs, manifoldsand cell complexes and potential theory. ... Read more

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In this extensive work, the authors offer a thorough introduction to multidimensional integral transformations. They discuss all of the most important transformatins, providing in a single volume a compilation of information that is at present scattered throughout the literature. Attention is concentrated on the operational calculus of the major integral transformatins and some of its applications, with an investigation of transforms in spaces of functions and of distributions. ... Read more

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Various related ideas are explored in this introductory survey, but no topic is treated thoroughly, since monographs are available on each topic separately. The Laplace transformation serves as a model for the developments of the operational properties of several other transformations. Further, it provides a connection to the Mikusinski calculus and generalized functions. The development of the Fourier transformation is followed by a different treatment of generalized functions (distributions). These introductions to generalized functions parallel the constructive approaches to the rational and real number systems. There are more examples than proofs, including applications in which the computations do not overwhelm the ideas. A calculus background and a scattering of other mathematical ideas is needed for self study. Deeper or lengthier problems are inserted along with simpler exercises. Several short tables of useful transform pairs are included.Audience: Upper level undergraduates and beginning graduate students in mathematics, physics and engineering. ... Read more

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Proceedings of the joint German-Israeli Workshop held March 1-10, 1995 in Tel Aviv, Israel. Contains a selection of papers in modern operator theory and its applications. ... Read more

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This book is devoted to the measure of non-compactness (essential norm) in weighted Lebesgue spaces for maximal, potential and singular operators dened, generally speaking, on homogeneous groups. The main topics of the monograph contain related results for potential and singular integrals in weighted function spaces with non-standard growth. One of the main characteristic features of the monograph is that the problems are studied in the two-weighted setting and cover the case of non-linear maps, such as, Hardy-Littlewood and fractional maximal functions. Before, these problems were investigated only for the restricted class of kernel operators consisting only of Hardy-type and Riemann-Liouville transforms. The book may be considered as a systematic and detailed analysis of a class of specific integral operators from the boundedness/compactness or non-compactness point of view. The material is self-contained and can be read by those with some background in real and functional analysis. ... Read more

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This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for mathematicians, applied scientists, and engineers.

The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others.

Innovative at every step, J. N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions.

The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn.

Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology.

The book includes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science.

The Hilbert transform made accessible with many new formulas and definitions

Written by today's foremost expert on the Hilbert transform of generalized functions, this combined text and reference covers the Hilbert transform of distributions and the space of periodic distributions. The author provides a consistently accessible treatment of this advanced-level subject and teaches techniques that can be easily applied to theoretical and physical problems encountered by mathematicians, applied scientists, and graduate students in mathematics and engineering.

Introducing many new inversion formulas that have been developed and applied by the author and his research associates, the book:

• Provides solutions to the distributional Hilbert problem and singular integral equations
• Focuses on the Hilbert transform of Schwartz distributions, giving intrinsic definitions of the space H(D) and its topology
• Covers the Paley-Wiener theorem and provides many important theoretical results of importance to research mathematicians
• Provides the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn
• Offers a new definition of the Hilbert transform of the periodic function that can be used for computational purposes in signal processing
• Develops the theory of the Hilbert transform of periodic distributions and the approximate Hilbert transform of periodic distributions
• Provides exercises at the end of each chapter—useful to professors in planning assignments, tests, and problems

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This book represents over twenty years of research ondistortions of functionals under actions of linear integral operators.It is divided into two parts. The first part addresses linear integral operators, establishing theirproperties and attempting to arrive at both specialisations as well asgeneralisations to be used in the second part. The second part isdevoted mainly to the development of several kinds of distortionsunder actions of integral operators for various familiar functionals.Among the topics that are treated are absolute modulus, real part,range, length and area, angular derivative, etc. Also, distortions onthe class of univalent functions and its subclasses,Caratheodory class, and distortions by a differential operatorare dealt with. Audience: This volume will be of interest to researchers andgraduate students whose work involves complex analysis, integraltransforms, operational calculus and operator theory. The volume isself-contained, giving many proofs, and is suitable for private studyas well as for seminars. ... Read more

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This volume provides a comprehensive, up-to-date survey of inequalities that involve a relationship between a function and its derivatives or integrals. The book is divided into 18 chapters, some of which are devoted to specific inequalities such as those of Kolmogorov-Landau, Wirtinger, Hardy, Carlson, Hilbert, Caplygin, Lyapunov, Gronwell and others. Over 800 references to the literature are cited; proofs are given when these provide insight into the general methods involved; and applications, especially to the theory of differential equations, are mentioned when appropriate. This volume will interest all those whose work involves differential and integral equations. It can also be recommended as a supplementary text. ... Read more