In this study, using the notion of statistical convergence, we obtain various statistical approximation theorems for a general sequence of max-product approximating operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical rates of the approximation. | Turk J Math 34 (2010) , 501 – 514. ¨ ITAK ˙ c TUB doi: Statistical convergence of max-product approximating operators Oktay Duman Abstract In this study, using the notion of statistical convergence, we obtain various statistical approximation theorems for a general sequence of max-product approximating operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical rates of the approximation. Key Words: Statistical convergence, max-product operators, Shepard operators, statistical rates. 1. Introduction In the classical approximation theory, many well-known approximating operators obey the linearity condition. In recent years, Bede et al. [3] have shown that it is possible to find some approximating operators that are not linear, such as, max-product and max-min Shepard type approximating operators. Actually, these operators are pseudo-linear which is a quite effective structure in solving the problems in many branches of applied mathematics, such as, image processing [4], differential equations [19, 20], idempotent analysis [18] and approximation theory [3, 5]. However, so far, almost all results regarding approximations by pseudolinear operators are based on the validity of the classical limit of the operators. Hence, in this paper, we focus on the following problem: is it possible to make an approximation by max-product operators although its classical limit fails? As an answer to this problem we mainly use the concept of statistical convergence, which was first introduced by Fast [13]. Recent studies demonstrate that the notion of statistical convergence provides an important contribution to the improvement of the classical approximation theory (see, for instance, [1, 2, 7, 8, 9, 10, 11, 12]). This paper is organized as follows: The first section is devoted to basic definitions and notations used in the paper. In the second section, we obtain some statistical approximation .
