Chebyshev Polynomials

Author:   J.C. Mason ,  David C. Handscomb (Oxford University Computing Laboratory, England, UK)
Publisher:   Taylor & Francis Ltd
ISBN:  

9780849303555


Pages:   360
Publication Date:   17 September 2002
Format:   Hardback
Availability:   In Print   Availability explained
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Chebyshev Polynomials


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Overview

Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject's state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebyshev polynomials lead one on a journey through all areas of numerical analysis. This book is the ideal vehicle with which to begin this journey and one that will also serve as a standard reference for many years to come.

Full Product Details

Author:   J.C. Mason ,  David C. Handscomb (Oxford University Computing Laboratory, England, UK)
Publisher:   Taylor & Francis Ltd
Imprint:   Chapman & Hall/CRC
Dimensions:   Width: 15.60cm , Height: 2.40cm , Length: 23.50cm
Weight:   0.635kg
ISBN:  

9780849303555


ISBN 10:   0849303559
Pages:   360
Publication Date:   17 September 2002
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

DEFINITIONS Preliminary Remarks Trigonometric Definitions and Recurrences Shifted Chebyshev Polynomials Chebyshev Polynomials of a Complex Variable BASIC PROPERTIES AND FORMULAE Introduction Chebyshev Polynomial Zeros and Extrema Chebyshev Polynomials and Power of x Evaluation of Chebyshev Sums, Products, Integrals and Derivatives MINIMAX PROPERTIES AND ITS APPLICATIONS Approximation-Theory and Structure Best and Minimax Approximation The Minimax Property of the Chebyshev Polynomials The Chebyshev Semi-Iterative Method for Linear Equations Telescoping Procedures for Power Series The Tau Method for Series and Rational Functions ORTHOGONALITY AND LEAST-SQUARES APPROXIMATION Introduction-From Minimax to Least Squares Orthogonality of Chebyshev Polynomials Orthogonal Polynomials and Best L2 Approximations Recurrence Relations Rodriques' Formulae and Differential Equations Discrete Orthogonality of Chebyshev Polynomials Discrete Chebyshev Transforms and the FFT Discrete Data Fitting by Orthogonal Polynomials-The Forsythe-Clenshaw Methods Orthogonality in the Complex Plane CHEBYSHEV SERIES Introduction-Chebyshev Series and Other Expansions Some Explicit Chebyshev Series Expansions Fourier-Chebyshev Series and Fourier Theory Projections and Near-Best Approximations Near-Minimax Approximation by a Chebyshev Series Comparison of Chebyshev and Other Polynomial Expansions The Error of a Truncated Chebyshev Expansion Series of Second-, Third-, and Fourth-Kind Polynomials Lacunary Chebyshev Series Chebyshev Series in the Complex Domain CHEBYSHEV INTERPOLATION Polynomial Interpolation Orthogonal Interpolation Chebyshev Interpolation Formulae Best L1 Approximation by Chebyshev Interpolation Near-Minimax Approximation by Chebyshev Interpolation NEAR-BEST L8, L1, and Lp APPROXIMATIONS Near-Best L8 (Near-Minimax) Approximations Near-Best L1 Approximations Best and Near-Best Lp Approximations INTEGRATION USING CHEBYSHEV POLYNOMIALS Indefinite Integration with Chebyshev Series Gauss-Chebyshev Quadrature Quadrature Methods of Clenshaw-Curtis Type Error Estimation for Clenshaw-Curtis Methods Some other Work on Clenshaw-Curtis Methods SOLUTION OF INTEGRAL EQUATIONS Introduction Fredholm Equations of the Second Kind Fredholm Equations of the Third Kind Fredholm Equations of the First Kind Singular Kernels Regularisation of Integral Equations Partial Differential Equations and Boundary Integral Equation Methods SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Introduction A Simple Example The Original Lanczos Tau Method A More General Linear Equation Pseudospectral Methods-Another Form of Collocation Nonlinear Equations Eigenvalue Problems Differential Equations in One Space and One Time Dimension CHEBYSHEV AND SPECTRAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS Introduction Interior, Boundary, and Mixed Methods Differentiation Matrices and Nodal Representation Method of Weighted Residuals Chebyshev Series and Galerkin Methods Collocation/Interpolation and Related Methods PDE Methods Some PDE Problems and Various Methods Computational Fluid Dynamics Particular Issues in Spectral Methods More Advanced Problems CONCLUSION BIBLIOGRAPHY APPENDICES Biographical Note Summary of Notations, Definitions, and Important Properties Tables of Coefficients INDEX Each chapter also contains a section of Problems.

Reviews

The book presents a wide panorama of the applications of Chebyshev polynomials to scientific computing. [It] is very clearly written and is a pleasure to read. Examples inserted in the text allow one to test his or her ability to understand and use the methods, which are described in detail, and each chapter ends with a section full of very pedagogical problems. Mathematics of Computation The book, by two well known specialists, is well written and presented. Many examples are given and problems have been added for students. It is a book that every numerical analyst should have. - Numerical Algorithms a modern treatment of the subject in a carefully prepared way very well produced - Mathematical Reviews, Issue 2004h


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