Towards Intelligent Modeling: Statistical Approximation Theory (Intelligent Systems Reference Library (14))

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. Front Matter....Pages - Introduction....Pages 1-8 Statistical Approximation by Bivariate Picard Singular Integral Operators....Pages 9-23 Uniform Approximation in Statistical Sense by Bivariate Gauss-Weierstrass Singular Integral Operators....Pages 25-38 Statistical L p -Convergence of Bivariate Smooth Picard Singular Integral Operators....Pages 39-60 Statistical L p -Approximation by Bivariate Gauss-Weierstrass Singular Integral Operators....Pages 61-83 A Baskakov-Type Generalization of Statistical Approximation Theory....Pages 85-96 Weighted Approximation in Statistical Sense to Derivatives of Functions....Pages 97-107 Statistical Approximation to Periodic Functions by a General Family of Linear Operators....Pages 109-115 Relaxing the Positivity Condition of Linear Operators in Statistical Korovkin Theory....Pages 117-129 Statistical Approximation Theory for Stochastic Processes....Pages 131-142 Statistical Approximation Theory for Multivariate Stochastic Processes....Pages 143-155 Fractional Korovkin-Type Approximation Theory Based on Statistical Convergence....Pages 157-167 Fractional Trigonometric Korovkin Theory Based on Statistical Convergence....Pages 169-180 Statistical Fuzzy Approximation Theory by Fuzzy Positive Linear Operators....Pages 181-188 Statistical Fuzzy Trigonometric Korovkin-Type Approximation Theory....Pages 189-197 High Order Statistical Fuzzy Korovkin-Type Approximation Theory....Pages 199-206 Statistical Approximation by Bivariate Complex Picard Integral Operators....Pages 207-215 Statistical Approximation by Bivariate Complex Gauss-Weierstrass Integral Operators....Pages 217-224 Back Matter....Pages -