George G. Lorentz, Manfred V. Golitschek,and Yuly Makovoz,Constructive Approximationα AdvancedProblems
✍ Scribed by Arno Kuijlaars
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 158 KB
- Volume
- 90
- Category
- Article
- ISSN
- 0021-9045
No coin nor oath required. For personal study only.
✦ Synopsis
Approximation theory is a diverse area, and this book clearly shows the width and depth of the subject. As a sequel to the monograph of R. A. DeVore's and G. G. Lorentz's ``Constructive Approximation'' (Vol. 303 of the same series, see the book review in J. Approx. Theory 78 (1994), 466 467), which presented a clear and systematic introduction to the approximation of functions of one real variable, this second volume, with partially different authors, treats a large number of additional questions in more detail. The range of topics presented is amazing and shows the rich diversity of the theory. Everyone working in approximation theory will be able to find something close to his area of interest. As a consequence of the diverse nature of the present volume, it is less coherent than its predecessor. Most of the 17 chapters can be read independently from one another.
Chapters 1 4 deal with polynomial approximations. Some of the many topics are the distribution of zeros and alternation points of best approximations, constrained polynomial approximation, approximation by incomplete polynomials, and approximation by weighted polynomials with varying weights. One of the highlights is a self-contained proof of a result of Lubinsky and Saff on weighted approximation by Freud weights.
Chapters 5 and 6 are about wavelets and splines. The chapter on wavelets presents the basic idea of multiresolution analysis and includes a recent construction, due to R. A. Lorentz and Sahakian, of orthonormal Schauder bases with trigonometric polynomials of low degree. The chapter on splines is a continuation of the discussion in the first volume. Splines of best approximation, periodic splines and the Schoenberg spline operator are treated.
Chapters 7 10 cover rational approximation. The authors discuss rational approximation for individual functions like e x on [&1, 1] and Popov's results for rational approximation of function classes. A separate chapter is devoted to Stahl's remarkable theorem on the error of best uniform rational approximation of |x| on [&1, 1]. The short chapter on Pade approximation includes the Nuttall Pommerenke theorem on convergence in capacity. This chapter is somewhat outside the main scope of the book, since it deals essentially with complex approximation. The same can be said about Chapter 10, which was prepared by Pekarskii and is based on his work on Hardy space methods for the error in best approximation.
The discussion of Mu ntz polynomials in Chapter 11 includes Jackson theorems and Markov-type inequalities. Some topics in nonlinear approximation are treated in Chapter 12. One finds Rice's theory of varisolvent families and abstract approximation in Banach spaces.
Chapters 13 15 discuss widths and entropies of classes of functions. Important results that are treated are the calculation of the n-widths of Sobolev classes, including recent results of Buslaev and Tikhomirov, and the asymptotics for n-widths of Lipschitz classes due to Kashin, Maiorov, Ho llig, and others. Based on results in arbitrary Banach spaces, the metric entropies of unit balls in Lipschitz spaces and spaces of analytic functions are estimated.