# tailieunhanh - Korovkin type approximation theorem for functions of two variables in statistical sense

## In this paper, using the concept of A-statistical convergence for double sequences, we investigate a Korovkin-type approximation theorem for sequences of positive linear operator on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. | Turk J Math 34 (2010) , 73 – 83. ¨ ITAK ˙ c TUB doi: Korovkin type approximation theorem for functions of two variables in statistical sense Fadime Dirik and Kamil Demirci Abstract In this paper, using the concept of A -statistical convergence for double sequences, we investigate a Korovkin-type approximation theorem for sequences of positive linear operator on the space of all continuous real valued functions deﬁned on any compact subset of the real two-dimensional space. Then we display an application which shows that our new result is stronger than its classical version. We also obtain a Voronovskaya-type theorem and some diﬀerential properties for sequences of positive linear operators constructed by means of the Bernstein polynomials of two variables. Key Words: A -Statistical convergence of double sequence, Korovkin-type approximation theorem, Bernstein polynomials, Voronovskaya-type theorem. 1. Introduction Let {Ln } be a sequence of positive linear operators acting from C(X) into C(X), which is the space of all continuous real valued functions on a compact subset X of all the real numbers. In this case, Korovkin [9] ﬁrst noticed necessary and suﬃcient conditions for the uniform convergence of Ln (f) to a function f by using the test functions ei deﬁned by ei (x) = xi (i = 0, 1, 2). Later many researchers investigate these conditions for various operators deﬁned on diﬀerent spaces. Furthermore, in recent years, with the help of the concept of statistical convergence, various statistical approximation results have been proved ([1], [2], [3], [4], [5], [8]). Recall that every convergent sequence (in the usual sense) is statistically convergent but its converse is not always true. Also, statistical convergent sequences do need to be bounded. So, the usage of this method of convergence in the approximation theory provides us many advantages. Our primary interest in the present paper is to obtain a Korovkin-type approximation theorem for

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