Four -dimensional Matrix Transformation and a Statistical Fuzzy Korovkin Type Approximation

In this paper, we prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators by using A-statistical convergence for four-dimensional summability matrices. Also we obtain rates of A-statistical convergence of a double sequence of fuzzy positive linear operators for fourdimensional summability matrices. 1. Introduction Anastassiou [3] …rst introduced the fuzzy analogue of the classical Korovkin theory (see also [1], [2], [4], [10]). Recently, some statistical fuzzy approximation theorems have been obtain by using the concept of statistical convergence (see, [5], [8]). In this paper, we prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators by using A-statistical convergence for four-dimensional summability matrices. Then, we construct an example such that our new approximation result works but its classical case does not work. Also we obtain rates of A-statistical convergence of a double sequence of fuzzy positive linear operators for four-dimensional summability matrices. We now recall some basic de…nitions and notations used in the paper. A fuzzy number is a function : R ! [0; 1], which is normal, convex, upper semi-continuous and the closure of the set supp( ) is compact, where supp( ) := fx 2 R : (x) > 0g. The set of all fuzzy numbers are denoted by RF . Let [ ] 0 = fx 2 R : (x) > 0g and [ ] = fx 2 R : (x) > rg , (0 < r < 1) : Then, it is well-known [11] that, for each r 2 [0; 1], the set [ ]is a closed and bounded interval of R. For any u; v 2 RF and 2 R, it is possible to de…ne uniquely the sum u v and the product u as follows: [u v] = [u] + [v] and [ u] = [u] , (0 r 1) : Now denote the interval [u] by h u (r) ; u (r) + i , where u u (r) + and u (r) ; u (r) + 2 R for r 2 [0; 1]. Then, for u; v 2 RF , de…ne u v , u v (r) and u (r) + v (r) + for all 0 r 1. De…ne also the following metric D : RF RF ! R+ by D(u; v) = sup r2[0;1] max n u v ; u + v + o : Key words and phrases. A statistical convergence for double sequences, fuzzy positive linear operators, fuzzy Korovkin theory, rates of A statistical convergence for double sequences, regularity for double sequences. 2010 Mathematics Subject Classi…cation. 26E50, 41A25, 41A36, 40G15 . 1 2 KAMIL DEMIRCI AND SEVDA KARAKUŞ Hence, (RF ; D) is a complete metric space [18]. A double sequence x = fxm;ng; m; n 2 N, is convergent in Pringsheim’s sense if, for every " > 0; there exists N = N(") 2 N such that jxm;n Lj < " whenever m;n > N . Then, L is called the Pringsheim limit of x and is denoted by P limm;n xm;n = L (see [16]). In this case, we say that x = fxm;ng is “P -convergent to L”. Also, if there exists a positive number M such that jxm;nj M for all (m;n) 2 N = N N; then x = fxm;ng is said to be bounded. Note that in contrast to the case for single sequences, a convergent double sequence need not to be bounded. A double sequence x = fxm;ng is said to be non-increasing in Pringsheim’s sense if, for all (m;n) 2 N; xm+1;n+1 xm;n. Now let A = [aj;k;m;n]; j; k;m; n 2 N, be a four-dimensional summability matrix. For a given double sequence x = fxm;ng, the A-transform of x, denoted by Ax := f(Ax)j;kg, is given by