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Frames (Vector analysis). --- Harmonic analysis. --- Time-series analysis. --- Benedetto, John Joseph,

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Operator theory --- 517.518 --- Frames (Vector analysis) --- Representations of groups --- Wavelets (Mathematics) --- Wavelet analysis --- Harmonic analysis --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Functional analysis --- Frame theory (Vector analysis) --- Vector analysis --- Metric theory of functions --- 517.518 Metric theory of functions --- Frames (mathématiques) --- Operator theory. --- Opérateurs, Théorie des --- Ondelettes --- Representations of groups. --- Représentations de groupes --- Opérateurs, Théorie des. --- Ondelettes. --- Représentations de groupes.

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John J. Benedetto has had a profound influence not only on the direction of harmonic analysis and its applications, but also on the entire community of people involved in the field. This self-contained volume in honor of John covers a wide range of topics in harmonic analysis and related areas, including weighted-norm inequalities, frame theory, wavelet theory, time-frequency analysis, and sampling theory. The invited chapters pay tribute to John’s many achievements and express an appreciation for both the mathematical and personal inspiration he has given to so many students, coauthors, and colleagues. 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 of harmonic analysis. The book is divided into the following five sections: * Classical harmonic analysis * Frame theory * Time-frequency analysis * Wavelet theory * Sampling theory and shift-invariant spaces Harmonic Analysis and Applications is an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, engineering, and physics. Contributors: A. Aldroubi, L. Baggett, G. Benke, C. Cabrelli, P.G. Casazza, O. Christensen, W. Czaja, M. Fickus, J.-P. Gabardo, K. Gröchenig, K. Guo, E. Hayashi, C. Heil, H.P. Heinig, J.A. Hogan, E. Kovacevic, D. Labate, J.D. Lakey, D. Larson, M.T. Leon, S. Li, W.-Q Lim, A. Lindner, U. Molter, A.M. Powell, B. Rom, E. Schulz, T. Sorrells, D. Speegle, K.F. Taylor, J.C. Tremain, D. Walnut, G. Weiss, E. Wilson, G. Zimmermann .

Harmonic analysis. --- Time-series analysis. --- Frames (Vector analysis) --- Benedetto, John. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Frame theory (Vector analysis) --- Vector analysis --- Analysis of time series --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Fourier analysis. --- Functional analysis. --- Operator theory. --- Mathematics. --- Abstract Harmonic Analysis. --- Fourier Analysis. --- Functional Analysis. --- Operator Theory. --- Approximations and Expansions. --- Math --- Science --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Approximation theory. --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems

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This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems.  Compared with the first edition, more emphasis is put on explicit constructions with attractive properties.  Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field.   Key features include: *Elementary introduction to frame theory in finite-dimensional spaces * Basic results presented in an accessible way for both pure and applied mathematicians * Extensive exercises make the work suitable as a textbook for use in graduate courses * Full proofs includ ed in introductory chapters; only basic knowledge of functional analysis required * Explicit constructions of frames and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems * Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory  * Selected research topics presented with recommendations for more advanced topics and further readin g * Open problems to stimulate further research   An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.   Review of the first edition: "Ole Christensen’s An Introduction to Frames and Riesz Bases is a first-rate introduction to the field … . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research."   — Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005 .

Mathematics. --- Harmonic analysis. --- Functional analysis. --- Operator theory. --- Functional Analysis. --- Abstract Harmonic Analysis. --- Operator Theory. --- Signal, Image and Speech Processing. --- Functional calculus --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Math --- Functional analysis --- Calculus of variations --- Functional equations --- Integral equations --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Science --- Frames (Vector analysis) --- Signal processing. --- Image processing. --- Speech processing systems. --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication)

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Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics.  More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas.  The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject. Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications.  With carefully selected contributions written by highly experienced researchers, it covers topics including: * Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames. Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory. With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more.  It is designed to be used as a supplemental textbook, self-study guide, or reference book.

Mathematics --- Operator theory --- Harmonic analysis. Fourier analysis --- Computer science --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- computervisie --- beeldverwerking --- Fourieranalyse --- analyse (wiskunde) --- toegepaste wiskunde --- grafische vormgeving --- informatica --- wiskunde --- KI (kunstmatige intelligentie) --- signaalverwerking --- Frames (Vector analysis). --- Finite frame theory. --- Mathematics. --- Fourier analysis. --- Computer vision. --- Operator theory. --- Approximations and Expansions. --- Signal, Image and Speech Processing. --- Fourier Analysis. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Operator Theory. --- Applications of Mathematics. --- Functional analysis --- Machine vision --- Vision, Computer --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Approximation theory. --- Signal processing. --- Image processing. --- Speech processing systems. --- Optical data processing. --- Applied mathematics. --- Engineering mathematics. --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Engineering --- Engineering analysis --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Optical equipment --- Finite frame theory --- Frames (Vector analysis)