Numerical Fourier Analysis

Author : Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche
Publisher : Springer
Page : 618 pages
File Size : 52,8 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 : Jayakumar Ramanathan
Publisher : Springer Science & Business Media
Page : 334 pages
File Size : 46,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461217565

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Methods of Applied Fourier Analysis by Jayakumar Ramanathan Pdf

Author : R. Vichnevetsky,J. B. Bowles
Publisher : SIAM
Page : 146 pages
File Size : 44,9 Mb
Release : 1982-01-01
Category : Technology & Engineering
ISBN : 9780898713923

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Fourier Analysis of Numerical Approximations of Hyperbolic Equations by R. Vichnevetsky,J. B. Bowles Pdf

This book provides useful reference material for those concerned with the use of Fourier analysis and computational fluid dynamics.

Author : Bao Luong
Publisher : Springer Science & Business Media
Page : 167 pages
File Size : 53,6 Mb
Release : 2009-08-14
Category : Mathematics
ISBN : 9780817649166

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Fourier Analysis on Finite Abelian Groups by Bao Luong Pdf

This unified, self-contained book examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups. With countless examples and unique exercise sets at the end of each section, Fourier Analysis on Finite Abelian Groups is a perfect companion to a first course in Fourier analysis. This text introduces mathematics students to subjects that are within their reach, but it also has powerful applications that may appeal to advanced researchers and mathematicians. The only prerequisites necessary are group theory, linear algebra, and complex analysis.

Author : M. W. Wong
Publisher : Springer Science & Business Media
Page : 177 pages
File Size : 42,6 Mb
Release : 2011-05-30
Category : Mathematics
ISBN : 9783034801164

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Discrete Fourier Analysis by M. W. Wong Pdf

This textbook presents basic notions and techniques of Fourier analysis in discrete settings. Written in a concise style, it is interlaced with remarks, discussions and motivations from signal analysis. The first part is dedicated to topics related to the Fourier transform, including discrete time-frequency analysis and discrete wavelet analysis. Basic knowledge of linear algebra and calculus is the only prerequisite. The second part is built on Hilbert spaces and Fourier series and culminates in a section on pseudo-differential operators, providing a lucid introduction to this advanced topic in analysis. Some measure theory language is used, although most of this part is accessible to students familiar with an undergraduate course in real analysis. Discrete Fourier Analysis is aimed at advanced undergraduate and graduate students in mathematics and applied mathematics. Enhanced with exercises, it will be an excellent resource for the classroom as well as for self-study.

Author : Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli
Publisher : Cambridge University Press
Page : 589 pages
File Size : 50,6 Mb
Release : 2018-06-21
Category : Mathematics
ISBN : 9781107182332

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Discrete Harmonic Analysis by Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli Pdf

A self-contained introduction to discrete harmonic analysis with an emphasis on the Discrete and Fast Fourier Transforms.

Author : Richard Bernatz
Publisher : John Wiley & Sons
Page : 336 pages
File Size : 54,5 Mb
Release : 2010-07-30
Category : Mathematics
ISBN : 9780470651377

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Fourier Series and Numerical Methods for Partial Differential Equations by Richard Bernatz Pdf

The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical methods that are essential for working with partial differential equations. Combining methodologies from calculus, introductory linear algebra, and ordinary differential equations (ODEs), the book strengthens and extends readers' knowledge of the power of linear spaces and linear transformations for purposes of understanding and solving a wide range of PDEs. The book begins with an introduction to the general terminology and topics related to PDEs, including the notion of initial and boundary value problems and also various solution techniques. Subsequent chapters explore: The solution process for Sturm-Liouville boundary value ODE problems and a Fourier series representation of the solution of initial boundary value problems in PDEs The concept of completeness, which introduces readers to Hilbert spaces The application of Laplace transforms and Duhamel's theorem to solve time-dependent boundary conditions The finite element method, using finite dimensional subspaces The finite analytic method with applications of the Fourier series methodology to linear version of non-linear PDEs Throughout the book, the author incorporates his own class-tested material, ensuring an accessible and easy-to-follow presentation that helps readers connect presented objectives with relevant applications to their own work. Maple is used throughout to solve many exercises, and a related Web site features Maple worksheets for readers to use when working with the book's one- and multi-dimensional problems. Fourier Series and Numerical Methods for Partial Differential Equations is an ideal book for courses on applied mathematics and partial differential equations at the upper-undergraduate and graduate levels. It is also a reliable resource for researchers and practitioners in the fields of mathematics, science, and engineering who work with mathematical modeling of physical phenomena, including diffusion and wave aspects.

Author : Ken'iti Kido
Publisher : Springer
Page : 210 pages
File Size : 40,5 Mb
Release : 2014-06-26
Category : Science
ISBN : 9781461492603

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Digital Fourier Analysis: Fundamentals by Ken'iti Kido Pdf

This textbook is a thorough, accessible introduction to digital Fourier analysis for undergraduate students in the sciences. Beginning with the principles of sine/cosine decomposition, the reader walks through the principles of discrete Fourier analysis before reaching the cornerstone of signal processing: the Fast Fourier Transform. Saturated with clear, coherent illustrations, "Digital Fourier Analysis" includes practice problems and thorough Appendices for the advanced reader. As a special feature, the book includes interactive applets (available online) that mirror the illustrations. These user-friendly applets animate concepts interactively, allowing the user to experiment with the underlying mathematics. For example, a real sine signal can be treated as a sum of clockwise and counter-clockwise rotating vectors. The applet illustration included with the book animates the rotating vectors and the resulting sine signal. By changing parameters such as amplitude and frequency, the reader can test various cases and view the results until they fully understand the principle. Additionally, the applet source code in Visual Basic is provided online, allowing this book to be used for teaching simple programming techniques. A complete, intuitive guide to the basics, "Digital Fourier Analysis - Fundamentals" is an essential reference for undergraduate students in science and engineering.

Author : Julius O. Smith
Publisher : Julius Smith
Page : 323 pages
File Size : 47,9 Mb
Release : 2008
Category : Fourier transformations
ISBN : 9780974560748

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Mathematics of the Discrete Fourier Transform (DFT) by Julius O. Smith Pdf

"The DFT can be understood as a numerical approximation to the Fourier transform. However, the DFT has its own exact Fourier theory, and that is the focus of this book. The DFT is normally encountered as the Fast Fourier Transform (FFT)--a high-speed algorithm for computing the DFT. The FFT is used extensively in a wide range of digital signal processing applications, including spectrum analysis, high-speed convolution (linear filtering), filter banks, signal detection and estimation, system identification, audio compression (such as MPEG-II AAC), spectral modeling sound synthesis, and many others. In this book, certain topics in digital audio signal processing are introduced as example applications of the DFT"--Back cover

Author : Claude Gasquet,Patrick Witomski
Publisher : Springer Science & Business Media
Page : 434 pages
File Size : 41,7 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461215981

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Fourier Analysis and Applications by Claude Gasquet,Patrick Witomski Pdf

The object of this book is two-fold -- on the one hand it conveys to mathematical readers a rigorous presentation and exploration of the important applications of analysis leading to numerical calculations. On the other hand, it presents physics readers with a body of theory in which the well-known formulae find their justification. The basic study of fundamental notions, such as Lebesgue integration and theory of distribution, allow the establishment of the following areas: Fourier analysis and convolution Filters and signal analysis time-frequency analysis (gabor transforms and wavelets). The whole is rounded off with a large number of exercises as well as selected worked-out solutions.

Author : Karlheinz Gröchenig
Publisher : Springer Science & Business Media
Page : 367 pages
File Size : 55,9 Mb
Release : 2013-12-01
Category : Technology & Engineering
ISBN : 9781461200031

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Foundations of Time-Frequency Analysis by Karlheinz Gröchenig Pdf

Time-frequency analysis is a modern branch of harmonic analysis. It com prises all those parts of mathematics and its applications that use the struc ture of translations and modulations (or time-frequency shifts) for the anal ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.

Author : Luca Brandolini
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 49,6 Mb
Release : 2004-08-06
Category : Mathematics
ISBN : 0817632638

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Fourier Analysis and Convexity by Luca Brandolini Pdf

Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians

Author : Rafael G. Campos
Publisher : Springer
Page : 235 pages
File Size : 49,7 Mb
Release : 2019-05-24
Category : Mathematics
ISBN : 9783030134235

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The XFT Quadrature in Discrete Fourier Analysis by Rafael G. Campos Pdf

This book has two main objectives, the first of which is to extend the power of numerical Fourier analysis and to show by means of theoretical examples and numerous concrete applications that when computing discrete Fourier transforms of periodic and non periodic functions, the usual kernel matrix of the Fourier transform, the discrete Fourier transform (DFT), should be replaced by another kernel matrix, the eXtended Fourier transform (XFT), since the XFT matrix appears as a convergent quadrature of a more general transform, the fractional Fourier transform. In turn, the book’s second goal is to present the XFT matrix as a finite-dimensional transformation that links certain discrete operators in the same way that the corresponding continuous operators are related by the Fourier transform, and to show that the XFT matrix accordingly generates sequences of matrix operators that represent continuum operators, and which allow these operators to be studied from another perspective.

Author : Gerlind Plonka,Daniel Potts,Gabriele Steidl,Manfred Tasche
Publisher : Springer Nature
Page : 676 pages
File Size : 44,6 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 : Jon H. Davis
Publisher : Springer Science & Business Media
Page : 744 pages
File Size : 49,8 Mb
Release : 2004
Category : Mathematics
ISBN : 0817643311

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Methods of Applied Mathematics with a MATLAB Overview by Jon H. Davis Pdf

Broadly organized around the applications of Fourier analysis, "Methods of Applied Mathematics with a MATLAB Overview" covers both classical applications in partial differential equations and boundary value problems, as well as the concepts and methods associated to the Laplace, Fourier, and discrete transforms. Transform inversion problems are also examined, along with the necessary background in complex variables. A final chapter treats wavelets, short-time Fourier analysis, and geometrically-based transforms. The computer program MATLAB is emphasized throughout, and an introduction to MATLAB is provided in an appendix. Rich in examples, illustrations, and exercises of varying difficulty, this text can be used for a one- or two-semester course and is ideal for students in pure and applied mathematics, physics, and engineering.