Discrete approximation theory (Q2826585)

In the book under review, the authors collect up-to-date and important results on the discrete singular operators. The book includes many mathematical proof techniques and ideas and it may serve as one of the most useful reference books in this field. Therefore it can be recommended to graduate students and postgraduate researchers in mathematics, physics and engineering. The book contains a rather rich bibliography. It is divided into eleven chapters as follows.NEWLINENEWLINENEWLINEThe first chapter deals with degree of uniform approximation by generalized discrete singular operators. Considering the discrete version of Gauss-Weierstrass operator introduced by Favard, the authors define a discrete version of it and study quantitatively their uniform approximation properties regarding convergence to the unit. The second chapter deals with degree of \(L_p\) approximation by generalized discrete singular operators. The quantitative \(L_p\) approximation properties of Picard, Gauss-Weierstrass and Poisson-Cauchy generalized discrete singular operators are studied. The third chapter deals with Voronovskaya like asymptotic expansions for these generalized discrete singular operators. The fourth chapter deals with the degree of approximation by fractional generalized discrete singular operators. The univariate and non-univariate cases, regarding their convergence to the unit operator with fractional rates in the uniform norm are studied. The fifth chapter deals with global smoothness preservation and simultaneous approximation by generalized discrete singular operators. The sixth chapter deals with approximation by complex generalized discrete singular operators. The seventh chapter deals with degree of uniform approximation by multivariate generalized discrete singular operators. The unitary and non-unitary cases of the operators are considered. Using multivariate higher order modulus of smoothness, the pointwise and uniform convergence of the operators to the unit operator are discussed. The \(L_p\) approximation properties of the operators are studied in Chapter 8, the Voronovskaya like asymptotic expansions for the operators are presented in Chapter 9, global smoothness preservation and simultaneous approximation by the operators are studied in Chapter 10 and in the Chapter 11 degree of approximation by multivariate complex generalized discrete singular operators are obtained.