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Multivariate polynomial approximation. (English) Zbl 1038.41002

ISNM. International Series of Numerical Mathematics 144. Basel: Birkhäuser (ISBN 3-7643-1638-1/hbk). x, 358 p. (2003).
Multivariate polynomials are an essential tool in approximation provided the complexity problem is mastered by a biorthogonal system or a kernel function basis. In the view of the recent development in this field, the author has written this interesting monograph which can be considered as extension of his former book [Constructive theory of multivariate functions: with an application to tomography (1990; Zbl 0717.41009)]. This book is divided into 4 parts. Part I has an introductory character and presents reproducing kernels, rotation principles, and Gegenbauer polynomials. Part II is devoted to approximation means, where polynomials on the sphere and ball as well as biorthogonal systems are discussed. The crucial part of this book is Part III on multivariate polynomial approximation. Important facts on best approximation, interpolation, quadrature, orthogonal projections and their summation are treated under a constructive view and embedded in the theory of positive linear operators. On this background, Part III gives also the first comprehensive introduction to the generalized hyperinterpolation. The hyperinterpolation was suggested by I. H. Sloan [J. Approximation Theory 83, 238–254 (1995; Zbl 0839.41006)] and later generalized by the author [Constructive Approximation 18, 183–203 (2002; Zbl 1002.41016)]. Generalized hyperinterpolation is a positive discrete polynomial approximation method, combining reasonable cost and uniform convergence, in particular cases at the best possible approximation order. First this theory is established on the sphere under an intrinsic asymptotic investigation of the node and weight distribution of positive quadratures. Then it is carried over to the balls of lower dimension by an identification of certain Laplace and Appell series. As an application, the Radon transform and the reconstruction problem of tomography are investigated in Part IV. In analogy to generalized hyperinterpolation, a polynomial reconstruction method with best possible convergence order is derived.
Each chapter closes with some problems. The solutions of all problems are given in the appendix. This well written book is of great interest to anyone working in multivariate polynomial approximation.

MSC:

41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
41A10 Approximation by polynomials
41A55 Approximate quadratures
41A63 Multidimensional problems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
44A12 Radon transform
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