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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.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
In this chapter, we mainly collect all necessary materials used in this book and discuss their fundamental properties. We also give some brief descriptions of the chapters.
George A. Anastassiou, Oktay Duman

Statistical Approximation by Bivariate Picard Singular Integral Operators

Abstract
At first we construct a sequence of bivariate smooth Picard singular integral operators which do not have to be positive in general. After giving some useful estimates, we mainly prove that it is possible to approximate a function by these operators in statistical sense even though they do not obey the positivity condition of the statistical Korovkin theory. This chapter relies on [25].
George A. Anastassiou, Oktay Duman

Uniform Approximation in Statistical Sense by Bivariate Gauss-Weierstrass Singular Integral Operators

Abstract
In this chapter, we study the statistical approximation properties of a sequence of bivariate smooth Gauss-Weierstrass singular integral operators which are not positive in general. We also show that the statistical approximation results are stronger than the classical uniform approximations. This chapter relies on [28].
George A. Anastassiou, Oktay Duman

Statistical L p -Convergence of Bivariate Smooth Picard Singular Integral Operators

Abstract
In this chapter, we obtain some statistical approximation results for the bivariate smooth Picard singular integral operators defined on L p -spaces, which do not need to be positive in general. Also, giving a non-trivial example we show that the statistical L p -approximation is stronger than the ordinary one. This chapter relies on [29].
George A. Anastassiou, Oktay Duman

Statistical L p -Approximation by Bivariate Gauss-Weierstrass Singular Integral Operators

Abstract
In this chapter, we study statistical L p -approximation properties of the bivariate Gauss-Weierstrass singular integral operators which are not positive in general. Furthermore, we introduce a non-trivial example showing that the statistical L p -approximation is more powerful than the ordinary case. This chapter relies on [23].
George A. Anastassiou, Oktay Duman

A Baskakov-Type Generalization of Statistical Approximation Theory

Abstract
In this chapter, with the help of the notion of A-statistical convergence, we get some statistical variants of Baskakov’s results on the Korovkin-type approximation theorems. This chapter relies on [16].
George A. Anastassiou, Oktay Duman

Weighted Approximation in Statistical Sense to Derivatives of Functions

Abstract
In this chapter, we prove some Korovkin-type approximation theorems providing the statistical weighted convergence to derivatives of functions by means of a class of linear operators acting on weighted spaces. We also discuss the contribution of these results to the approximation theory. This chapter relies on [19].
George A. Anastassiou, Oktay Duman

Statistical Approximation to Periodic Functions by a General Family of Linear Operators

Abstract
In this chapter, using A-statistical convergence and also considering some matrix summability methods, we introduce an approximation theorem, which is a non-trivial generalization of Baskakov’s result [40] regarding the approximation to periodic functions by a general class of linear operators. This chapter relies on [20].
George A. Anastassiou, Oktay Duman

Relaxing the Positivity Condition of Linear Operators in Statistical Korovkin Theory

Abstract
In this chapter, we relax the positivity condition of linear operators in the Korovkin-type approximation theory via the concept of statistical convergence. Especially, we prove some Korovkin-type approximation theorems providing the statistical convergence to derivatives of functions by means of a class of linear operators. This chapter relies on [18].
George A. Anastassiou, Oktay Duman

Statistical Approximation Theory for Stochastic Processes

Abstract
In this chapter, we present strong Korovkin-type approximation theorems for stochastic processes via the concept of A-statistical convergence. This chapter relies on [31].
George A. Anastassiou, Oktay Duman

Statistical Approximation Theory for Multivariate Stochastic Processes

Abstract
In this chapter, we obtain some Korovkin-type approximation theorems for multivariate stochastic processes with the help of the concept of A-statistical convergence. A non-trivial example showing the importance of this method of approximation is also introduced. This chapter relies on [26].
George A. Anastassiou, Oktay Duman

Fractional Korovkin-Type Approximation Theory Based on Statistical Convergence

Abstract
In this chapter, we get some statistical Korovkin-type approximation theorems including fractional derivatives of functions. Furthermore, we demonstrate that these results are more applicable than the classical ones. This chapter relies on [21].
George A. Anastassiou, Oktay Duman

Fractional Trigonometric Korovkin Theory Based on Statistical Convergence

Abstract
In this chapter, we develop the classical trigonometric Korovkin theory by using the concept of statistical convergence from the summability theory and also by considering the fractional derivatives of trigonometric functions. We also show that these results are more applicable than the classical ones. This chapter relies on [27].
George A. Anastassiou, Oktay Duman

Statistical Fuzzy Approximation Theory by Fuzzy Positive Linear Operators

Abstract
In this chapter, we give a Korovkin-type approximation theorem for fuzzy positive linear operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix. This type of approximation enables us to obtain more powerful results than in the classical aspects of approximation theory settings. An application of this result is also presented. Furthermore, we study the rates of this statistical fuzzy convergence of the operators via the fuzzy modulus of continuity. This chapter relies on [17].
George A. Anastassiou, Oktay Duman

Statistical Fuzzy Trigonometric Korovkin-Type Approximation Theory

Abstract
In this chapter, we consider non-negative regular summability matrix transformations in the approximation by fuzzy positive linear operators, where the test functions are trigonometric. So, we mainly obtain a trigonometric fuzzy Korovkin theorem by means of A-statistical convergence. We also compute the rates of A-statistical convergence of a sequence of fuzzy positive linear operators in the trigonometric environment. This chapter relies on [59].
George A. Anastassiou, Oktay Duman

High Order Statistical Fuzzy Korovkin-Type Approximation Theory

Abstract
In this chapter, we obtain a statistical fuzzy Korovkin-type approximation result with high rate of convergence. Main tools used in this work are statistical convergence and higher order continuously differentiable functions in the fuzzy sense. An application is also given, which demonstrates that the statistical fuzzy approximation is stronger than the classical one. This chapter relies on [22].
George A. Anastassiou, Oktay Duman

Statistical Approximation by Bivariate Complex Picard Integral Operators

Abstract
In this chapter, we investigate some statistical approximation properties of the bivariate complex Picard integral operators. Furthermore, we show that the statistical approach is more applicable than the well-known aspects. This chapter relies on [24].
George A. Anastassiou, Oktay Duman

Statistical Approximation by Bivariate Complex Gauss-Weierstrass Integral Operators

Abstract
In this chapter, we present the complex Gauss-Weierstrass integral operators defined on a space of analytic functions in two variables on the Cartesian product of two unit disks. Then, we investigate some geometric properties and statistical approximation process of these operators. This chapter relies on [30].
George A. Anastassiou, Oktay Duman

Backmatter

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