Applied and Numerical Harmonic Analysis – serie

Over the course of the last century, the systematic exploration of the relationship between Fourier analysis and other branches of mathematics has lead to important advances in geometry, number theory, and analysis, stimulated in part by Hurwitz’s proof of the isoperimetric inequality using Fourier series. This unified, self-contained book presents both a broad overview of Fourier analysis and convexity, as well as an intricate look at applications in some specific settings; it will be useful to graduate students and researchers in harmonic analysis, convex geometry, functional analysis, number theory, computer science, and combinatorial analysis. A wide audience will benefit from the careful demonstration of how Fourier analysis is used to distill the essence of many mathematical problems in a natural and elegant way.

This concisely written book gives an elementary introduction to a classical area of mathematics - approximation theory - in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Approximation Theory: From Taylor Polynomials to Wavelets will be an excellent textbook or self-study reference for students and instructors in pure and applied mathematics, mathematical physics, and engineering. Readers will find motivation and background material pointing toward advanced literature and research topics in pure and applied harmonic analysis and related areas.

Advances in Discrete Tomography and its Applications

This volume, in honor of John J. Benedetto on the occasion of his 65th birthday, features invited articles covering a wide range of topics in areas where he has made fundamental and lasting contributions, such as harmonic analysis, number theory, weighted norm inequalities, wavelet theory, time-frequency analysis, and sampling theory.Although the scope of the book is broad, chapters are clustered by topic to provide authoritative expositions that will be of lasting interest. The original papers collected here are written by prominent, well-respected researchers and professionals in the field. "Harmonic Analysis and its Applications" pays tribute to Benedetto's many achievements and expresses an appreciation for both the mathematical and personal inspiration that he has given to so many students, coauthors, and colleagues.Contributors: A. Aldroubi, L. Baggett, G. Benke, C.A. Cabrelli, P.G. Casazza, O. Christensen, W. Czaja, C. Davis, H.G. Feichtinger, M. Fickus, J.-P. Gabardo, K. Groechenig, K. Guo, E. Hayashi, C. Heil, H.P. Heinig, J. Hogan, P.E.T. Jorgensen, K. Kornelson, J. Kovacevic, D. Labate, J.D. Lakey, D.R. Larson, M.T. Leon, S. Li, W.-Q. Lim, A. Linder, U.M. Molter, E.Schulz, T. Sorrels, D. Speegle, K.F. Taylor, J.C. Tremain, D.F. Walnut, Y. Wang, G. Weiss, E. Wilson.

Concerned with the mathematical theory of surfaces, objects and their boundaries in discrete spaces, this text provides a self-contained and mathematically precise introduction to the field. It is application-oriented and presents problems of visualization and analysis of multidimensional data sets. The primary areas of mathematics used are graph theory and topology.

This resource examines both theoretical and practical aspects of computational signal processing using wavelets. Computationally, wavelet signal processing algorithms are presented and applied to signal compression, noise supression, and signal identification. Numerical illustrations of these computational techniques are further discussed in the text (using MATLAB) and the software M-Files are available via the World Wide Web site for the book. Starting from basic principles of signal representation with atomic functions, a mathematically well-founded theory of the discretization of analogue signals is developed. General families are specialized to wavelet families, with discrete representation specialized to generally non-orthogonal wavelet transforms. The theory leads naturally to the computer implementation of the non-orthagonal wavelet transform. Specific topics covered include general signal representation, continuous wavelet transform, multi-resolution analysis, continuous wavelet transform, non-orthagonal wavelet transform, and wavelet based signal processing algorithms for compression, noise supression, and identification.The technical discussion is at the begninning graduate level and is accessible to all signal processing professionals and practitioners.

This text presents a theory of time-frequency representations over finite and finitely generated abelian groups which can be used to design algorithms for multidimensional applications in imaging, electromagnetics and communication theory. Emphasis is placed on Weyl-Heisenberg systems and expansions. Algorithms are developed within this abstract setting without reference to co-ordinates or dimension. By not concerning itself with co-ordinates and dimensions, algorithmic structures can be derived which should be of importance to multidimensional applications in mathematics and electrical engineering.

Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems which is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practioners and researchers. The goal of the book is to give the reader, specialist and non-specialist useable and modern mathematical tools in their research and analysis. This new text is intended for graduate students and researchers in applied mathematics, physical sciences and engineering. The careful explanations, accessible writing style, and many illustrations/examples also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise.

In 1994, in my role as Technical Program Chair for the 17th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, I solicited proposals for mini-symposia to provide delegates with accessible summaries of important issues in research areas outside their particular specializations. Terry Peters and his colleagues submitted a proposal for a symposium on Fourier Trans­ forms and Biomedical Engineering whose goal was "to demystify the Fourier transform and describe its practical application in biomedi­ cal situations". This was to be achieved by presenting the concepts in straightforward, physical terms with examples drawn for the parti­ cipants work in physiological signal analysis and medical imaging. The mini-symposia proved to be a great success and drew a large and appreciative audience. The only complaint being that the time allocated, 90 minutes, was not adequate to allow the participants to elaborate their ideas adequately. I understand that this feedback helped the authors to develop this book.

Distributions in the Physical and Engineering Sciences, Volume 2

This text provides an introduction to mathematicians and engineers who want to learn about the different approaches and aspects of Gabor analysis or want to apply Gabor-based techniques to tasks in signal and image processing. It should be of interest to those working in the fields of harmonic analysis, applied mathematics, numerical analysis, engineering, signal and image processing, optics and pattern recognition. Features include: self-contained introduction to basic Gabor analysis; survey of fundamental results in Gabor theory; efficient numerical algorithms; Gabor expansions in signal and image processing; and applications in pattern recognition, filter bank design and optics.

A new text/reference offering a comprehensive and detailed presentation of wavelet theory, principles and methods. It presents basic theory of wavelet bases and transforms without assuming knowledge of advanced mathematics. The material is presented with many examples, exercises and thorough references. An essential text/reference for applied mathematicians, engineers and scientists. The goal of this book is to present the basics of wavelet theory in a complete, rigorous, and cogent fashion, but at a level that is truly appropriate for upper level undergraduates, first year graduate students, or anyone who has had a course in advanced calculus. The book presents the basic theory of wavelet bases and transforms without assuming any knowledge of Lebesgue integration or the theory of abstract Hilbert spaces. The book motivates the central ideas of wavelet theory by doing a detailed exposition of Haar series, and then shows how a more abstract approach allows us to generalize and improve upon Haar series. Once these ideas have been established and explored, variations and extensions of the Haar construction are presented.Applications of the theory are to be stressed throughout the book as a means of motivating the usefulness of the theory, but a more complete treatment of applications will be saved for the last three chapters of the book. The book will also include exercises at the end of each chapter.

This volume presents a development of the ideas of harmonic analysis with a special emphasis on application-oriented themes. In keeping with the interdisciplinary nature of the subject, theoretical aspects of the subject are complemented by in-depth explorations of related material of an applied nature. Thus, basic material on Fourier series, Hardy spaces and the Fourier transform are interwoven with chapters treating the discrete Fourier transform and fast algorithms, the spectral theory of stationary processes, H-infinity control theory, and wavelet theory.

At the heart of partial differential equations is an unchanging core of material, but the subject is constantly expanding and evolving. This advanced textbook reference is an introduction to partial differential equations covering the traditional subjects of the heat equation, wave equation and Laplace equation in the traditional manner using the method of separation of variables.

This work presents a systematic introduction to time-frequency analysis understood as a central area of applied harmonic analysis, while at the same time honouring its interdisciplinary origins. Important principles include: classic Fourier analysis as a tool that is central in modern mathematics; the mathematical structures based on the operations of translation and modulations (i.e: the Heisenberg group); the many forms of the uncertainty principle; and the omnipresence of Gaussian functions, both in the methodology of proofs and in important statements.

A state-of-the-art edited survey covering all aspects of sampling theory. Theory, methods and applications are discussed in authoritative expositions ranging from multi-dimensional signal analysis to wavelet transforms. The book is an essential up-to-date resource for all researchers and professionals in applied math and engineering working in fourier analysis, wavelets, signal processing and time-frequency analysis. Sampling is a fundamental topic in the engineering and physical sciences. This new edited book focuses on recent mathematical methods and theoretical developments, as well as some current central applications of the Classical Sampling Theorem. The Classical Sampling Theorem, which originated in the 19th century, is often associated with the names of Shannon, Kotelnikov, and Whittaker; and one of the features of this book is an English translation of the pioneering work in the 1930s by Kotelnikov, a Russian engineer. Following a technical overview and Kotelnikov's article, the book includes a wide and coherent range of mathematical ideas essential for modern sampling techniques.These ideas involve wavelets and frames, complex and abstract harmonic analysis, the Fast Fourier Transform (FFT), and special functions and eigenfunction expansions. Some of the applications addressed are tomography and medical imaging.

Methods of signal analysis represent a broad research topic with applications in many disciplines, including engineering, technology, biomedicine, seismography, eco­ nometrics, and many others based upon the processing of observed variables. Even though these applications are widely different, the mathematical background be­ hind them is similar and includes the use of the discrete Fourier transform and z-transform for signal analysis, and both linear and non-linear methods for signal identification, modelling, prediction, segmentation, and classification. These meth­ ods are in many cases closely related to optimization problems, statistical methods, and artificial neural networks. This book incorporates a collection of research papers based upon selected contri­ butions presented at the First European Conference on Signal Analysis and Predic­ tion (ECSAP-97) in Prague, Czech Republic, held June 24-27, 1997 at the Strahov Monastery. Even though the Conference was intended as a European Conference, at first initiated by the European Association for Signal Processing (EURASIP), it was very gratifying that it also drew significant support from other important scientific societies, including the lEE, Signal Processing Society of IEEE, and the Acoustical Society of America. The organizing committee was pleased that the re­ sponse from the academic community to participate at this Conference was very large; 128 summaries written by 242 authors from 36 countries were received. In addition, the Conference qualified under the Continuing Professional Development Scheme to provide PD units for participants and contributors.

Goals of the Book Overthelast thirty yearsthere has been arevolutionindiagnostic radiology as a result oftheemergenceofcomputerized tomography (CT), which is the process of obtaining the density distribution within the human body from multiple x-ray projections. Since an enormous variety of possible density values may occur in the body, a large number of projections are necessary to ensure the accurate reconstruction oftheir distribution. There are other situations in which we desire to reconstruct an object from its projections, but in which we know that the object to be recon­ structed has only a small number of possible values. For example, a large fraction of objects scanned in industrial CT (for the purpose of nonde­ structive testing or reverse engineering) are made of a single material and so the ideal reconstruction should contain only two values: zero for air and the value associated with the material composing the object. Similar as­ sumptions may even be made for some specific medical applications; for example, in angiography ofthe heart chambers the value is either zero (in­ dicating the absence of dye) or the value associated with the dye in the chamber. Another example arises in the electron microscopy of biological macromolecules, where we may assume that the object to be reconstructed is composed of ice, protein, and RNA. One can also apply electron mi­ croscopy to determine the presenceor absence ofatoms in crystallinestruc­ tures, which is again a two-valued situation.

The last fifteen years have produced major advances in the mathematical theory of wavelet transforms and their applications to science and engineering. In an effort to inform researchers in mathematics, physics, statistics, computer science, and engineering and to stimulate furtherresearch, an NSF-CBMS Research Conference on Wavelet Analysis was organized at the University of Central Florida in May 1998. Many distinguished mathematicians and scientists from allover the world participated in the conference and provided a digest of recent developments, open questions, and unsolved problems in this rapidly growing and important field. As a follow-up project, this monograph was developed from manuscripts sub­ mitted by renowned mathematicians and scientists who have made important contributions to the subject of wavelets, wavelet transforms, and time-frequency signal analysis. This publication brings together current developments in the theory and applications of wavelet transforms and in the field of time-frequency signal analysis that are likely to determine fruitful directions for future advanced study and research.

This self-contained book provides a basic foundation for students, practitioners, and researchers interested in some of the diverse new areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled and analyzed using a continuous flow of observations from land or satellite devices. Harmonic wavelet methods are introduced, as well as fast computational schemes and various numerical test examples.Topic and key features: - Comprehensive coverage of topics which, thus far, are only scattered in journal articles and conference proceedings- Important applications and developments for future satellite scenarios; new modelling techniques involving low-orbiting satellites- Multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling- Exercises at the end of each chapter and an appendix with hints to their solutions

Wavelets and Subbands: Fundamentals and Applications is an essential text reference for graduates, researchers, and professionals in electrical engineering, communications engineering, and computer engineering. Practitioners and professionals engaged in signal processing, wavelets and fourier analysis will find the book a useful resource and comprehensive guide.

Provides a digest of the current developments, open questions and unsolved problems likely to determine a new frontier for future advanced study and research in the rapidly growing areas of wavelets, wavelet transforms, signal analysis, and signal and image processing. Key topics include multivariate wavelets, wavelet transforms, time-frequency signal analysis, self-similarity and intermittency problems in turbulence, wavelet image compression, class of band-limited wavelets and multiresolution analysis. An essential text/reference for advanced students and practitioners in wavelets, and wavelet transforms, signal processing and time-frequency signal analysis. Professionals working in electrical and computer engineering, applied mathematics, computer science, biomedical engineering, physics, optics, and fluid mechanics will also find the book a valuable resource.

Gabor analysis constitutes a central part of time-frequency analysis. While preserving the symmetry between the time (location) domain and the frequency (wave number) domain, Gabor analysis avoids the high degree of redundancy inherent in the continuous short-time Fourier transform. The ability to resolve details of a signal (or a system impulse response) in a two-dimensional representation, whose coefficients have a very natural interpretation, is the basis for many interesting applications in electrical engineering, and signal-and image processing making Gabor analysis an important branch of applied mathematics. This work provides an overview of recent developments in Gabor analysis. Leading scientists in various disciplines related to the subject treat a range of topics from covering theory to numerics as well as applications of Gabor analysis. The presentation unfolds systematically and includes appropriate background material in each chapter, placing the work in its proper context. Graduate students, practitioners and researchers in the areas of numerical analysis, electrical engineering and applied mathematics will finds this book a current and useful resource.

Developed in this book are several deep connections between time-frequency (Fourier/Gabor) analysis and time-scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of developments in time-frequency and time-scale analysis are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time-frequency and time-scale methods as a rich area of work.

Transforms have diverse applications in digital signal processing and other areas of science, engineering, and technology. Indeed, new transforms are continuously emerging to solve many new or open problems created by technological advances. To improve the implementation of these transforms, fast algorithms have been created and widely used over the last forty years. Although closely related to the discrete Fourier transform, novel fast algorithms for various transforms have to be developed separately in order to minimize computational complexity and implementation costs. This book is a comprehensive presentation of results and recent progress on several commonly used transforms and their fast algorithms. In many cases, additional options are provided for new or improved fast algorithms, some not yet well known in the digital signal processing community.

? Concise background material for each chapter, open problems, exercises, bibliography, and comprehensive index make this work a fine pedagogical and reference resource.; New previously unpublished results appear on the homotopy of multiresolutions, approximation theory, the spectrum and structure of the fixed points of the associated transfer, subdivision operators; Key topics of wavelet theory are examined; Excellent graphics show how wavelets depend on the spectra of the transfer operators; The important role of the spectrum of a transfer operator is studied; This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory.

Sampling, wavelets, and tomography are three active areas of contemporary mathematics sharing common roots that lie at the heart of harmonic and Fourier analysis. The advent of new techniques in mathematical analysis has strengthened their interdependence and led to some new and interesting results in the field. This state-of-the-art book not only presents new results in these research areas, but it also demonstrates the role of sampling in both wavelet theory and tomography.Specific topics covered include:* Robustness of Regular Sampling in Sobolev Algebras * Irregular and Semi-Irregular Weyl-Heisenberg Frames * Adaptive Irregular Sampling in Meshfree Flow Simulation * Sampling Theorems for Non-Bandlimited Signals * Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets * Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces * Sampling Theory and Parallel-Beam Tomography * Thin-Plate Spline Interpolation in Medical Imaging * Filtered Back-Projection Algorithms for Spiral Cone Computed TomographyAimed at mathematicians, scientists, and engineers working in signal and image processing and medical imaging, the work is designed to be accessible to an audience with diverse mathematical backgrounds. Although the volume reflects the contributions of renowned mathematicians and engineers, each chapter has an expository introduction written for the non-specialist. One of the key features of the book is an introductory chapter stressing the interdependence of the three main areas covered. A comprehensive index completes the work. Contributors: J. J. Benedetto, N. K. Bose, P. G. Casazza, Y. C. Eldar, H. G.Feichtinger, A. Faridani, A. Iske, S. Jaffard, A. Katsevich, S. Lertrattanapanich, G. Lauritsch, B. Mair, M. Papadakis, P. P. Vaidyanathan, T. Werther, D. C. Wilson, A. I. Zayed

Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schrodinger's, Einstein's and Newton's equations. Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. The text is enriched with good examples and exercises at the end of every chapter. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Satellite navigation receivers are used to receive and decode satellite navigation signals, including those provided by the GPS constellation of satellites. There is an increasing need for a unified platform that will enable enhanced receiver development and design, as well as cost-effective testing procedures for various applications. This book explores the use of such new technologies in the area of satellite navigation receivers. In order to obtain a reconfigurable receiver with a wide range of applications, the authors discuss receiver architecture based on software-defined radio techniques. The presentation unfolds in a systematic user-friendly style and goes from the basics to cutting-edge research.Key features and topics include:* presentation of basic signal structures used in GPS and Galileo---the European satellite navigation system * design and implementation of a GPS signal generator using the simulated signals * analysis of three different methods of signal acquisition: the serial search, the parallel-frequency space search, and the newer method of parallel-code phase search * implementation of analyzed methods in MATLAB and a discussion of the choice of algorithms involved * MATLAB-based exercises * a hands-on method of testing the material covered in the book: hardware equipment---which may be purchased online---enables readers to generate real-world data, and downloadable MATLAB software---available at www gps.aau.dk/~borre/sw--- allows readers to change various parameters and immediately see their effect * bibliography of recent results and comprehensive index The book is aimed at applied mathematicians, electrical engineers, geodesists, and graduate students. It may be used as a textbook in various GPS technology and signal processing courses, or as a self-study reference for anyone working with satellite navigation receivers.