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Chebyshev polynomials. (English) Zbl 1015.33001
Boca Raton, FL: Chapman & Hall/CRC. xiv, 341 p. (2003).
This book constitutes an up-to-date treatment of the theory and applications of Chebyshev polynomials. It provides a rigorous and readable exposition of the state of art of the subject. The volume contains twelve chapters. Chapter 1 is devoted to the trigonometric definitions of the four kinds of Chebyshev polynomials, their recurrences and connections. Chebyshev polynomials of a complex variable are also considered. Chapter 2 deals with basic properties and formulae including zeros and extrema, relations with powers of \(x\) and evaluation of Chebyshev sums, products, integrals and derivatives. The minimax property of Chebyshev polynomials and its applications which include the acceleration of the convergence of iterative methods for linear equations, the telescoping procedures for power series and the tau method for series and rational functions are presented in Chapter 3. The other important property of the orthogonality of the four families of Chebyshev polynomials is described in Chapter 4 together with the least-squares continuous and discrete approximations.In Chapter 5 is pointed out the importance of Chebyshev series as an example of orthogonal polynomial expansion that may be transformed into a Fourier series or a Laurent series according to whether the independent variable is real or complex. Chapters 6 and 7 are concerned with Chebyshev polynomial interpolations and near-best \({\mathcal L}_\infty \), \({\mathcal L}_1 \), and \({\mathcal L}_p \) approximations, respectively. Chapter 8 illustrates the use of Chebyshev polynomials in numerical integration. First, the indefinite integration with Chebyshev series and the Gauss-Chebyshev quadrature are recalled, next the quadrature methods of Clenshaw-Curtis type of the four kinds are discussed in detail with their error estimation. Further, an interesting account of the recent literature on Clenshaw-Curtis methods is given. The last part of the book is devoted to the solution of integral equations, ordinary and partial differential equations which are now major fields of application for Chebyshev polynomials. In Chapter 9 particular emphasis is given to the regularization methods for integral equations, a smoothing algorithm with weighted function regularization and other recent results are described. Chapter 10 illustrates collocation, projection and pseudospectral methods for linear and nonlinear ordinary differential equations. Chapter 11, which deals with Chebyshev and spectral methods for partial differential equations, contains an interesting and of good didactic value development and description of spectral and pseudo-spectral methods by working through a selection of problems of progressively increasing complexity. Chapter 12, of only half-page, contains the conclusion. The book closes with a list of more than 200 references, three Appendices consisting in a biographical note, summary of notations, definitions and properties and tables of coefficients. The presentation of the material is very clear and supported by properly chosen problems added to each chapter. The book is a welcome addition to the literature on the numerical methods. It will be fully appreciated by the researchers in the area of numerical analysis and its applications.

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
41A50 Best approximation, Chebyshev systems