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.
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.
Biography
David C. Handscomb, J.C. Mason
"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
