A Basis Theory Primer

Expanded Edition

Author: Christopher Heil

Publisher: Springer Science & Business Media

ISBN: 0817646868

Category: Mathematics

Page: 534

View: 4862

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The classical subject of bases in Banach spaces has taken on a new life in the modern development of applied harmonic analysis. This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and its use in both applied and classical harmonic analysis. The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory. * Part I develops the functional analysis that underlies most of the concepts presented in the later parts of the text. * Part II presents the abstract theory of bases and frames in Banach and Hilbert spaces, including the classical topics of convergence, Schauder bases, biorthogonal systems, and unconditional bases, followed by the more recent topics of Riesz bases and frames in Hilbert spaces. * Part III relates bases and frames to applied harmonic analysis, including sampling theory, Gabor analysis, and wavelet theory. * Part IV deals with classical harmonic analysis and Fourier series, emphasizing the role played by bases, which is a different viewpoint from that taken in most discussions of Fourier series. Key features: * Self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications. * Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses; hints for selected exercises are included at the end of the book. * A separate solutions manual is available for instructors upon request at: www.birkhauser-science.com/978-0-8176-4686-8/. * No other text develops the ties between classical basis theory and its modern uses in applied harmonic analysis. A Basis Theory Primer is suitable for independent study or as the basis for a graduate-level course. Instructors have several options for building a course around the text depending on the level and background of their students.
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Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth

Author: Michael Cwikel,Mario Milman

Publisher: American Mathematical Soc.

ISBN: 1470428369

Category: Fourier analysis

Page: 411

View: 2657

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This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways. Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool. This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.
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Excursions in Harmonic Analysis, Volume 1

The February Fourier Talks at the Norbert Wiener Center

Author: Travis D Andrews,Radu Balan,John J. Benedetto,Wojciech Czaja,Kasso A. Okoudjou

Publisher: Springer Science & Business Media

ISBN: 0817683763

Category: Mathematics

Page: 488

View: 4498

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The Norbert Wiener Center for Harmonic Analysis and Applications provides a state-of-the-art research venue for the broad emerging area of mathematical engineering in the context of harmonic analysis. This two-volume set consists of contributions from speakers at the February Fourier Talks (FFT) from 2006-2011. The FFT are organized by the Norbert Wiener Center in the Department of Mathematics at the University of Maryland, College Park. These volumes span a large spectrum of harmonic analysis and its applications. They are divided into the following parts: Volume I · Sampling Theory · Remote Sensing · Mathematics of Data Processing · Applications of Data Processing Volume II · Measure Theory · Filtering · Operator Theory · Biomathematics Each part provides state-of-the-art results, with contributions from an impressive array of mathematicians, engineers, and scientists in academia, industry, and government. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center is an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, engineering, and physics.
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Harmonic Analysis: A Comprehensive Course in Analysis, Part 3

Author: Barry Simon

Publisher: American Mathematical Soc.

ISBN: 1470411024

Category: Harmonic analysis

Page: 759

View: 4711

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A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis. Part 3 returns to the themes of Part 1 by discussing pointwise limits (going beyond the usual focus on the Hardy-Littlewood maximal function by including ergodic theorems and martingale convergence), harmonic functions and potential theory, frames and wavelets, spaces (including bounded mean oscillation (BMO)) and, in the final chapter, lots of inequalities, including Sobolev spaces, Calderon-Zygmund estimates, and hypercontractive semigroups.
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An Introduction to Frames and Riesz Bases

Author: Ole Christensen

Publisher: Birkhäuser

ISBN: 3319256130

Category: Mathematics

Page: 704

View: 4858

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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
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Wavelet Theory and Harmonic Analysis in Applied Sciences

Author: Carlos Enrique D'Attellis,Elena M. Fernandez-Berdaguer

Publisher: Springer Science & Business Media

ISBN: 9780817639532

Category: Mathematics

Page: 345

View: 8690

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This book contains 14 invited chapters addressing applications and interactions between wavelet theory and scientific, medical and geophysical problems. Topics covered include EGG signals, spectral analysis, wavelet transform from orthogonal spline wavelets, numerical modeling of Maxwell's equation, wavelet networks and nonlinear processes, etc. It is addressed to an interdisciplinary readership of professional applied mathematicians, electrical engineers, physicists and other scientists especially interested in applying the new ideas and techniques. Series: Applied and Numerical Harmonic Analysis (ANHA) Contents I. Theory and Implementations 1. Singular integrals related to the Monge-Ampere equation -L.A.Caffarelli and C. Gutierrez 1. Introduction 2. The maximal function 3. Application to singular integral operators 4. References 2. Wavelet characterization of functions with conditions on the mean oscillation -H. Aimar and A. Bernardis 1. Introduction 2. Wavelet Bases 3. Functions with conditions on the mean oscillation: MO spaces 4. Sequential spaces of Carleson type: C spaces 5. C as a necessary condition for the wavelet coefficients of MO functions 5.1 The Daubechies wavelet case 5.2 The Meyer wavelet case 6. C as a sufficient condition for MO : The finite case 6.1 The Daubechies wavelet case 6.2 The Meyer wavelet case 7. C as sufficient condition for MO : The general case 8. Acknowledgments 9. References 3. Undecimated Wavelet Transform from Orthogonal Spline Wavelets -E.P. Serrano and M.A. Fabio 1. Introduction 2. The Undecimated Discrete Wavelet Transform 3. Undecimate Cardinal Algorithm 4. Properties of the Undecimated Discrete Transform 5. Extended Multiresolution Analysis for V 0 6. Decomposition of a Signal in V 0 7. Conclusion 8. References 4. Oblique Multiwavelet Bases -A. Aldroubi 1. Introduction 1.1 An oblique wavelet based on the Haar multiresolution 2. Multiscaling Functions and Multiwavelet Bases 2.1 Multiscaling functions 2.2 Multiwavelets 2.3 Classification of wavelet bases 2.4 Two-scale equations 3. Construction of Oblique Wavelet and Multiwavelet Bases 3.1 Oblique multiwavelet bases based on the Hermite cubic splines MRA 4. Fast Filter-Bank Algorithms 4.1 Initialization 5. References 5. Frames and Riesz bases: a short survey -S.J. Favier and R.A. Zalik 1. Introduction 2. Frames and Riesz bases in Hilbert Spaces 3. Frame and Basis Perturbations 4. Special Classes of Frames and The Proyection Method 5. Exponential Frames and Bases 6. Wavelet Frames, Bases, and Bessel Sequences 7. References 6. Fourier Analysis of Petrov-Galerkin Methods Based on Biorthogonal Multiresolution Analyses -S.M. Gomes and E. Cortina 1. Introduction 2. Notation and some definitions 3. A Petrov-Galerkin method for the KdV equation 4. The linear case: Fourier analysis 4.1 Convergence results 4.2 Conditions for stability 5. Biorthogonal framework 5.1 Spline biorthogonal scaling functions 5.2 Algorithms for the calculation of a(k), b(l,k) and c(k) 6. Numerical results 7. Acknowledgment 8. References II. Applications to Biomedical Sciences 7. Fine Structure of ECG Signal using Wavelet Transform -H. Rix and O. Meste 1. Introduction 2. Localization of Late Potentials by Time Frequency Representations 3. Signal Shape Differences 3.1 Shape classifications of 1-D signals 3.2 Shape classification of 2-D signals 4. Conclusion 5. References 8. Spectral Analysis of Cardiorespiratory Signals -M. Risk, J. Sobh, R. Armentano, A. Ramirez and P. Saul 1. Introduction 2. Short term variability 2.1 Spectral Analysis Methods 2.2 Spectral Analysis 2.3 Autoregressive and moving average modeling 2.4 Fast Fourier Transform method 2.5 Blackman-Tukey method 2.6 Transfer Function 3. Long term variability 3.1 Time domain methods 3.2 Frequency domain methods 4. Discussion 5. References 9. Characterization of Epileptic EEG Time Series (I): Gabor Transform and Nonlinear Dynamics Methods -S. Blanco, S. Kochen, R. Quian Quiroga, L. Riquelme, O. Rosso and P. Salgado 1. Introduction 2. Experimental Setup and Clinical Data 3. Armonic Analysis of EEG Data 4. Time-Frequency Analysis Based on Gabor Transform 4.1 Analysis of EEG Signal I 4.2 Analysis of EEG Signal II 4.3 Information Transfer Analysis 5. Nonlinear Dynamics Analysis 5.1 Stationarity 5.2 Dynamical Systems 5.3 Attractors 5.4 Attractor Reconstruction 5.5 Choosing the Optimal Time Delay 5.6 Choosing the Minimum Embedding Dimension 5.7 Correlation Dimension 5.8 Lyapunov Exponent 5.9 Analysis of Signal II 6. Final Remarks 7. Acknowledgements 8. References 10. Characterization of Epileptic EEG Time Series (II): Wavelet Transform and Information Theory -C. D'Attellis, L. Gamero, S. Isaacson, R. Sirne and M. Torres 1. Introduction 2. Data Collection 3. Theoretical Background 3.1 Wavelet Analysis and Filters 3.2 Entropy 4. Method I: Energy Based Detection Algorithm 4.1 Results and Comparisons 4.2 Discussion 5. Method II: Multiresolution Entropy 5.1 Results 5.2 Discussion 6. Conclusions 7. References III. Applications in Physical Sciences 11. Wavelet Networks for Modelling Nonlinear Processes -N. Roqueiro and E.L. Lima 1. Introduction 2. Models for non-linear system identification 2.1 Volterra's series 2.2 Block oriented models 2.3 Non-linear models linear in the parameters 3. Wavelet networks 3.1 Multivariable wavelet networks 3.2 Parallel structure of single variable wavelets 3.3 Conclusions 4. Examples 4.1 The non-linear identification problem 4.2 Identification of a chaotic attractor 4.3 Continuous stirred tank reactor identification 4.4 Conclusions 5. General conclusion 6. References 12. Higher order asymptotic boundary conditions for an oxide region in a semiconductor device -I. Gamba 1. Introduction 2. The full problem 2.1 Regularity of the full problem 2.2 Equivalent problem formulated in terms of the Fourier representation 2.3 Higher order asymptotic behavior of the Oxide boundary condition 2.4 Conclusions 3. References 13. Estimation of the complex plain-wave modulus in viscoelastic media -E.M. Fernandez-Berdaguer and J.E. Santos 1. Introduction 1.1 Description of the Viscoelastic Model 1.2 Formulation of the Estimation Problem 1.3 The Gateaux Derivatives 1.4 The Algorithms 1.5 The Adjoint Problem 1.6 Numerical Experiments 2. References 14. Numerical Modelling of Maxwell's Equations with Applications to Magnetotellurics -J.E. Santos and L. Guarracino 1. The Differential Model and the Interative Hybrid Finite Element Domain Decomposition Algorithm 1.1 The Differential Model 1.2 A Differential Domain Decomposition Formulation 1.3 The Iterative Hybrid Finite Element Domain Decomposition Procedure 1.4 Experimental Calculations 1.5 Some remarks on the implementation of the algorithm 2. Conclusions 3. References
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Theoretical Numerical Analysis

A Functional Analysis Framework

Author: Kendall Atkinson,Weimin Han

Publisher: Springer Science & Business Media

ISBN: 1441904581

Category: Mathematics

Page: 625

View: 2100

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This textbook prepares graduate students for research in numerical analysis/computational mathematics by giving to them a mathematical framework embedded in functional analysis and focused on numerical analysis. This helps the student to move rapidly into a research program. The text covers basic results of functional analysis, approximation theory, Fourier analysis and wavelets, iteration methods for nonlinear equations, finite difference methods, Sobolev spaces and weak formulations of boundary value problems, finite element methods, elliptic variational inequalities and their numerical solution, numerical methods for solving integral equations of the second kind, and boundary integral equations for planar regions. The presentation of each topic is meant to be an introduction with certain degree of depth. Comprehensive references on a particular topic are listed at the end of each chapter for further reading and study. Because of the relevance in solving real world problems, multivariable polynomials are playing an ever more important role in research and applications. In this third editon, a new chapter on this topic has been included and some major changes are made on two chapters from the previous edition. In addition, there are numerous minor changes throughout the entire text and new exercises are added. Review of earlier edition: "...the book is clearly written, quite pleasant to read, and contains a lot of important material; and the authors have done an excellent job at balancing theoretical developments, interesting examples and exercises, numerical experiments, and bibliographical references." R. Glowinski, SIAM Review, 2003
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Gabor Analysis and Algorithms

Theory and Applications

Author: Hans G. Feichtinger,Thomas Strohmer

Publisher: Springer Science & Business Media

ISBN: 1461220165

Category: Mathematics

Page: 496

View: 5644

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In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.
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Wavelets

Mathematics and Applications

Author: John J. Benedetto

Publisher: CRC Press

ISBN: 9780849382710

Category: Mathematics

Page: 592

View: 1458

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Wavelets is a carefully organized and edited collection of extended survey papers addressing key topics in the mathematical foundations and applications of wavelet theory. The first part of the book is devoted to the fundamentals of wavelet analysis. The construction of wavelet bases and the fast computation of the wavelet transform in both continuous and discrete settings is covered. The theory of frames, dilation equations, and local Fourier bases are also presented. The second part of the book discusses applications in signal analysis, while the third part covers operator analysis and partial differential equations. Each chapter in these sections provides an up-to-date introduction to such topics as sampling theory, probability and statistics, compression, numerical analysis, turbulence, operator theory, and harmonic analysis. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. It will be an especially useful reference for harmonic analysts, partial differential equation researchers, signal processing engineers, numerical analysts, fluids researchers, and applied mathematicians.
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A First Course in Wavelets with Fourier Analysis

Author: Albert Boggess,Francis J. Narcowich

Publisher: John Wiley & Sons

ISBN: 1118211154

Category: Mathematics

Page: 336

View: 7259

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A comprehensive, self-contained treatment of Fourier analysisand wavelets—now in a new edition Through expansive coverage and easy-to-follow explanations, AFirst Course in Wavelets with Fourier Analysis, SecondEdition provides a self-contained mathematical treatment of Fourieranalysis and wavelets, while uniquely presenting signal analysisapplications and problems. Essential and fundamental ideas arepresented in an effort to make the book accessible to a broadaudience, and, in addition, their applications to signal processingare kept at an elementary level. The book begins with an introduction to vector spaces, innerproduct spaces, and other preliminary topics in analysis.Subsequent chapters feature: The development of a Fourier series, Fourier transform, anddiscrete Fourier analysis Improved sections devoted to continuous wavelets andtwo-dimensional wavelets The analysis of Haar, Shannon, and linear spline wavelets The general theory of multi-resolution analysis Updated MATLAB code and expanded applications to signalprocessing The construction, smoothness, and computation of Daubechies'wavelets Advanced topics such as wavelets in higher dimensions,decomposition and reconstruction, and wavelet transform Applications to signal processing are provided throughout thebook, most involving the filtering and compression of signals fromaudio or video. Some of these applications are presented first inthe context of Fourier analysis and are later explored in thechapters on wavelets. New exercises introduce additionalapplications, and complete proofs accompany the discussion of eachpresented theory. Extensive appendices outline more advanced proofsand partial solutions to exercises as well as updated MATLABroutines that supplement the presented examples. A First Course in Wavelets with Fourier Analysis, SecondEdition is an excellent book for courses in mathematics andengineering at the upper-undergraduate and graduate levels. It isalso a valuable resource for mathematicians, signal processingengineers, and scientists who wish to learn about wavelet theoryand Fourier analysis on an elementary level.
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