Anastassiou G.A., Duman O. Towards Intelligent Modeling: Statistical Approximation Theory

Berlin: Springer, 2011. — 252 p.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.Statistical Approximation by Bivariate Picard Singular Integral Operators
Uniform Approximation in Statistical Sense by Bivariate Gauss-Weierstrass Singular Integral Operators
Statistical L p -Convergence of Bivariate Smooth Picard Singular Integral Operators
Statistical L p -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 Stochastic Processes
Fractional Korovkin-Type Approximation Theory Based on Statistical Convergence
Fractional Trigonometric Korovkin Theory Based on Statistical Convergence
Statistical Fuzzy Approximation Theory by Fuzzy Positive Linear Operators
Statistical Fuzzy Trigonometric Korovkin-Type Approximation Theory
High Order Statistical Fuzzy Korovkin-Type Approximation Theory
Statistical Approximation by Bivariate Complex Picard Integral Operators
Statistical Approximation by Bivariate Complex Gauss-Weierstrass Integral Operators