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Constructive approximation. (English) Zbl 0797.41016
Grundlehren der Mathematischen Wissenschaften. 303. Berlin: Springer- Verlag. x, 449 p. (1993).
This is a remarkable treatise on constructive approximation theory, written by two of the leading experts in the field. It contains an up to date systematic and unified exposition of some basic problems in approximation theory of functions of one real variable. As it is mentioned in the Preface, in some sense, this volume represents a modern version of the corresponding parts of the book published earlier by the second author: Approximation of functions. First edition: Holt, Rinehart and Winston, New York (1966; Zbl 0153.389), Second edition: Chelsea, New York (1986; Zbl 0643.41001). Great care was taken in the selection of material, which is well organized and carefully worked out.
This volume consists of 13 chapters. The chapter headings are as follows: (1) Theorems of Weierstrass; (2) Spaces of functions; (3) Best approximation; (4) Properties of polynomials; (5) Splines; (6) \(K\)- functionals and interpolation spaces; (7) Central theorems of approximation; (8) Influence of endpoints in polynomial approximation; (9) Approximation by operators; (10) Bernstein polynomials; (11) Approximation of classes of functions, Müntz theorems; (12) Spline approximation; (13) Spline interpolation and projections onto spline spaces.
The book closes with a bibliography extended over 13 pages and a subject index. We mention that each paper is supplied with references for the page or pages where it has been used. The authors try to give a complete exposition of the main basic theorems of the theory, treating the most general cases or discussing special problems.
Some of the results are due to the authors themselves, but they have done an excellent job of collecting and unifying other new results on the subject, obtained by other mathematicians. This history of the subject was not neglected.
The book contains also practical algorithms of approximation. Occasionally was used certain methods of functional analysis for establishing some theoretical general results. The authors succeeded to present in a masterful way the present state of knowledge of the constructive approximation theory. The problems and the notes attached to the chapters will be of a valuable use.
This book will be a useful source of information for researchers in constructive function theory and, in general, for mathematicians and engineers whose work involve approximation and numerical methods.
{Reviewer’s remarks: (i) At page 331, in the expression of the Meyer- König, and Zeller operator, was omitted the binomial coefficient \({{m+k} \choose k}\); (ii) As there has been pointed out by the reviewer in Math. Zeitschr. 98, 46-51 (1967; Zbl 0169.074), where was given a deduction of a formula and an extension of it, the monotonicity property of the sequence of Bernstein polynomials, for convex functions, was established first by W. B. Temple [Duke Math. J. 21, 527-531 (1954; Zbl 0058.050)]}.

MSC:
41A25 Rate of convergence, degree of approximation
41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
41A15 Spline approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
41A40 Saturation in approximation theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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