Synopses & Reviews
The main idea of statistical convergence is to demand convergence only for a majority of elements of a sequence. This method of convergence has been investigated in many fundamental areas of mathematics such as: measure theory, approximation theory, fuzzy logic theory, summability theory, and so on. In this monograph we consider this concept in approximating a function by linear operators, especially when the classical limit fails. The results of this book not only cover the classical and statistical approximation theory, but also are applied in the fuzzy logic via the fuzzy-valued operators. The authors in particular treat the important Korovkin approximation theory of positive linear operators in statistical and fuzzy sense. They also present various statistical approximation theorems for some specific real and complex-valued linear operators that are not positive. This is the first monograph in Statistical Approximation Theory and Fuzziness. The chapters are self-contained and several advanced courses can be taught. The research findings will be useful in various applications including applied and computational mathematics, stochastics, engineering, artificial intelligence, vision and machine learning. This monograph is directed to graduate students, researchers, practitioners and professors of all disciplines.
Review
From the reviews: "This is the first monograph in statistical approximation theory and fuzziness, which contains mostly the recent joint works of the authors ... . The book consists of eighteen chapters which are self-contained and include many significant applications. ... A complete list of references and a useful index are presented at the end of the book. This monograph is recommended to graduate students and researchers, in both pure and applied mathematics, specializing in summability and approximation theories." (Cihan Orhan, Mathematical Reviews, Issue 2012 k)
Review
From the reviews:
"This is the first monograph in statistical approximation theory and fuzziness, which contains mostly the recent joint works of the authors ... . The book consists of eighteen chapters which are self-contained and include many significant applications. ... A complete list of references and a useful index are presented at the end of the book. This monograph is recommended to graduate students and researchers, in both pure and applied mathematics, specializing in summability and approximation theories." (Cihan Orhan, Mathematical Reviews, Issue 2012 k)
Synopsis
Statistical convergence has been explored in numerous contexts such as fuzzy logic theory. Here, the authors approach the subject by approximating a function by linear operators, focusing on situations in which the classical limit is not effective.
Table of Contents
Introduction.-
Statistical Approximation by Bivariate Picard Singular Integral Operators.-
Uniform Approximation in Statistical Sense by Bivariate Gauss-Weierstrass Singular Integral Operators.-
Statistical Lp-Convergence of Bivariate Smooth Picard Singular Integral Operators.-
Statistical Lp-Approximation by Bivariate Gauss-Weierstrass Singular Integral Operators.-
A Baskakov-Type Generalization of Statistical Approximation Theory.-
Weighted Approximation in Statistical Sense to Derivatives of Functions.-
Statistical Approximation to Periodic Functions by a General Family of Linear Operators.-
Relaxing the Positivity Condition of Linear Operators in Statistical Korovkin Theory.-
Statistical Approximation Theory for Stochastic Processes.- Statistical Approximation Theory for Multivariate Stochas tic Processes.