Methods Of Fourier Analysis And Approximation Theory

Author : Michael Ruzhansky,Sergey Tikhonov
Publisher : Birkhäuser
Page : 258 pages
File Size : 51,5 Mb
Release : 2016-03-11
Category : Mathematics
ISBN : 9783319274669

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Methods of Fourier Analysis and Approximation Theory by Michael Ruzhansky,Sergey Tikhonov Pdf

Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.

Author : P.L. Butzer,Nessel,Trebels
Publisher : Birkhäuser
Page : 565 pages
File Size : 47,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034874489

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Fourier Analysis and Approximation by P.L. Butzer,Nessel,Trebels Pdf

At the international conference on 'Harmonic Analysis and Integral Transforms', conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans forms from a transform point of view, (ii) treatment of Fourier transforms in LP(lRn)_ space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolu tion type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the one-dimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.

Author : Roald M. Trigub,Eduard S. Belinsky
Publisher : Springer Science & Business Media
Page : 595 pages
File Size : 44,5 Mb
Release : 2012-11-07
Category : Mathematics
ISBN : 9781402028762

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Fourier Analysis and Approximation of Functions by Roald M. Trigub,Eduard S. Belinsky Pdf

In Fourier Analysis and Approximation of Functions basics of classical Fourier Analysis are given as well as those of approximation by polynomials, splines and entire functions of exponential type. In Chapter 1 which has an introductory nature, theorems on convergence, in that or another sense, of integral operators are given. In Chapter 2 basic properties of simple and multiple Fourier series are discussed, while in Chapter 3 those of Fourier integrals are studied. The first three chapters as well as partially Chapter 4 and classical Wiener, Bochner, Bernstein, Khintchin, and Beurling theorems in Chapter 6 might be interesting and available to all familiar with fundamentals of integration theory and elements of Complex Analysis and Operator Theory. Applied mathematicians interested in harmonic analysis and/or numerical methods based on ideas of Approximation Theory are among them. In Chapters 6-11 very recent results are sometimes given in certain directions. Many of these results have never appeared as a book or certain consistent part of a book and can be found only in periodics; looking for them in numerous journals might be quite onerous, thus this book may work as a reference source. The methods used in the book are those of classical analysis, Fourier Analysis in finite-dimensional Euclidean space Diophantine Analysis, and random choice.

Author : György Alexits,Paul Turán
Publisher : North-Holland
Page : 468 pages
File Size : 43,6 Mb
Release : 1978
Category : Mathematics
ISBN : UOM:39015028049982

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Fourier Analysis and Approximation Theory by György Alexits,Paul Turán Pdf

Author : Paul Butzer,Nessel,Trebels
Publisher : Birkhäuser
Page : 554 pages
File Size : 44,9 Mb
Release : 1971-01-01
Category : Mathematics
ISBN : 3764305207

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Fourier Analysis and Approximation by Paul Butzer,Nessel,Trebels Pdf

At the international conference on 'Harmonic Analysis and Integral Transforms', conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans forms from a transform point of view, (ii) treatment of Fourier transforms in LP(lRn)_ space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolu tion type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the one-dimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.

Author : Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche
Publisher : Springer Nature
Page : 676 pages
File Size : 51,7 Mb
Release : 2023-11-08
Category : Mathematics
ISBN : 9783031350054

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Numerical Fourier Analysis by Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche Pdf

New technological innovations and advances in research in areas such as spectroscopy, computer tomography, signal processing, and data analysis require a deep understanding of function approximation using Fourier methods. To address this growing need, this monograph combines mathematical theory and numerical algorithms to offer a unified and self-contained presentation of Fourier analysis. The first four chapters of the text serve as an introduction to classical Fourier analysis in the univariate and multivariate cases, including the discrete Fourier transforms, providing the necessary background for all further chapters. Next, chapters explore the construction and analysis of corresponding fast algorithms in the one- and multidimensional cases. The well-known fast Fourier transforms (FFTs) are discussed, as well as recent results on the construction of the nonequispaced FFTs, high-dimensional FFTs on special lattices, and sparse FFTs. An additional chapter is devoted to discrete trigonometric transforms and Chebyshev expansions. The final two chapters consider various applications of numerical Fourier methods for improved function approximation, including Prony methods for the recovery of structured functions. This new edition has been revised and updated throughout, featuring new material on a new Fourier approach to the ANOVA decomposition of high-dimensional trigonometric polynomials; new research results on the approximation errors of the nonequispaced fast Fourier transform based on special window functions; and the recently developed ESPIRA algorithm for recovery of exponential sums, among others. Numerical Fourier Analysis will be of interest to graduate students and researchers in applied mathematics, physics, computer science, engineering, and other areas where Fourier methods play an important role in applications.

Author : György Alexits,Paul Turán
Publisher : Unknown
Page : 476 pages
File Size : 42,9 Mb
Release : 1978
Category : Approximation theory
ISBN : UOM:39015037744045

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Fourier Analysis and Approximation Theory by György Alexits,Paul Turán Pdf

Author : Mark A. Pinsky
Publisher : American Mathematical Society
Page : 398 pages
File Size : 53,7 Mb
Release : 2023-12-21
Category : Mathematics
ISBN : 9781470475673

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Introduction to Fourier Analysis and Wavelets by Mark A. Pinsky Pdf

This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs–Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces $L^p(mathbb{R}^n)$. Chapter 4 gives a gentle introduction to these results, using the Riesz–Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry–Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the $L_2$ theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material.

Author : Paul Leo Butzer,Rolf Joachim Nessel
Publisher : Unknown
Page : 553 pages
File Size : 52,8 Mb
Release : 1971
Category : Mathematics
ISBN : 0121485013

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Fourier Analysis and Approximation by Paul Leo Butzer,Rolf Joachim Nessel Pdf

Author : Volker Michel
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 51,7 Mb
Release : 2012-12-12
Category : Mathematics
ISBN : 9780817684037

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Lectures on Constructive Approximation by Volker Michel Pdf

Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball focuses on spherical problems as they occur in the geosciences and medical imaging. It comprises the author’s lectures on classical approximation methods based on orthogonal polynomials and selected modern tools such as splines and wavelets. Methods for approximating functions on the real line are treated first, as they provide the foundations for the methods on the sphere and the ball and are useful for the analysis of time-dependent (spherical) problems. The author then examines the transfer of these spherical methods to problems on the ball, such as the modeling of the Earth’s or the brain’s interior. Specific topics covered include: * the advantages and disadvantages of Fourier, spline, and wavelet methods * theory and numerics of orthogonal polynomials on intervals, spheres, and balls * cubic splines and splines based on reproducing kernels * multiresolution analysis using wavelets and scaling functions This textbook is written for students in mathematics, physics, engineering, and the geosciences who have a basic background in analysis and linear algebra. The work may also be suitable as a self-study resource for researchers in the above-mentioned fields.

Author : Charles Jean de La Vallée Poussin
Publisher : Unknown
Page : 644 pages
File Size : 47,7 Mb
Release : 2004
Category : Mathematics
ISBN : UOM:39015059578586

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Approximation theory, Fourier analysis, quasi-analytic functions by Charles Jean de La Vallée Poussin Pdf

Author : Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche
Publisher : Springer
Page : 618 pages
File Size : 40,8 Mb
Release : 2019-02-05
Category : Mathematics
ISBN : 9783030043063

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Numerical Fourier Analysis by Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche Pdf

This book offers a unified presentation of Fourier theory and corresponding algorithms emerging from new developments in function approximation using Fourier methods. It starts with a detailed discussion of classical Fourier theory to enable readers to grasp the construction and analysis of advanced fast Fourier algorithms introduced in the second part, such as nonequispaced and sparse FFTs in higher dimensions. Lastly, it contains a selection of numerical applications, including recent research results on nonlinear function approximation by exponential sums. The code of most of the presented algorithms is available in the authors’ public domain software packages. Students and researchers alike benefit from this unified presentation of Fourier theory and corresponding algorithms.

Author : Andrei A. Gonchar,Edward B. Saff
Publisher : Springer
Page : 225 pages
File Size : 44,5 Mb
Release : 2008-01-03
Category : Mathematics
ISBN : 9783540477921

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Methods of Approximation Theory in Complex Analysis and Mathematical Physics by Andrei A. Gonchar,Edward B. Saff Pdf

The book incorporates research papers and surveys written by participants ofan International Scientific Programme on Approximation Theory jointly supervised by Institute for Constructive Mathematics of University of South Florida at Tampa, USA and the Euler International Mathematical Instituteat St. Petersburg, Russia. The aim of the Programme was to present new developments in Constructive Approximation Theory. The topics of the papers are: asymptotic behaviour of orthogonal polynomials, rational approximation of classical functions, quadrature formulas, theory of n-widths, nonlinear approximation in Hardy algebras,numerical results on best polynomial approximations, wavelet analysis. FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics for orthogonal polynomials associated with exponential weights on R.- A.L. Levin, E.B. Saff: Exact Convergence Rates for Best Lp Rational Approximation to the Signum Function and for Optimal Quadrature in Hp.- H. Stahl: Uniform Rational Approximation of x .- M. Rahman, S.K. Suslov: Classical Biorthogonal Rational Functions.- V.P. Havin, A. Presa Sague: Approximation properties of harmonic vector fields and differential forms.- O.G. Parfenov: Extremal problems for Blaschke products and N-widths.- A.J. Carpenter, R.S. Varga: Some Numerical Results on Best Uniform Polynomial Approximation of x on 0,1 .- J.S. Geronimo: Polynomials Orthogonal on the Unit Circle with Random Recurrence Coefficients.- S. Khrushchev: Parameters of orthogonal polynomials.- V.N. Temlyakov: The universality of the Fibonacci cubature formulas.

Author : Anonim
Publisher : Academic Press
Page : 554 pages
File Size : 41,6 Mb
Release : 2011-09-21
Category : Mathematics
ISBN : 0080873537

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Fourier Analysis and Approximation by Anonim Pdf

Fourier Analysis and Approximation

Author : Jayakumar Ramanathan
Publisher : Springer Science & Business Media
Page : 334 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461217565

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Methods of Applied Fourier Analysis by Jayakumar Ramanathan Pdf