On Multivariate Bleimann, Butzer and Hahn Operators
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On Multivariate Bleimann, Butzer and Hahn Operators Dilek S¨oylemez, Didem Aydın Arı and G¨ ulen Ba¸scanbaz-Tunca Abstract. In this paper, we state a Korovkin-type theorem for uniform approximation of functions, belonging to a class generated by multivariable function of modulus of continuity, by the sequence of multivariate positive linear operators. Then, using this theorem, we investigate the corresponding uniform approximation result for the multivariate Bleimann, Butzer and Hahn operators which are not in a tensor product design. Moreover, we give an elementary proof that these operators are non-increasing in n when the attached function is convex and non-increasing and we add a graphical example. Mathematics Subject Classification. 41A36, 41A63. Keywords. Multivariate Bleimann, Butzer and Hahn operator, total modulus of continuity, monotonicity.
1. Introduction Approximation theory is concerned with how functions can best be approximated with simpler functions. Since Korovkin’s theorem was obtained, the study of the linear methods of approximation given by sequences of linear positive operators has become an important area in approximation theory. Many authors have studied this theory with various motivation considering some special sequences of positive linear operators and their generalizations. For a detailed knowledge, we refer to the book of Altomare and Campiti [9]. In [12], Bleimann, Butzer and Hahn proposed a sequence of positive linear operators Ln defined by n k n k 1 Ln (f ; x) = x f , x ≥ 0, n ∈ N, (1.1) n k n+1−k (1 + x) k=0
and proved that Ln (f ; x) → f (x) as n → ∞ pointwise on [0, ∞) when f ∈ CB [0, ∞) := {f ∈ C[0, ∞) : f is bounded on [0, ∞)} , 0123456789().: V,-vol
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where C[0, ∞) denotes the space of continuous, real valued functions defined on [0, ∞). Moreover, convergence being uniform on each compact subset of [0, ∞). Jayasri and Sitaraman [17] studied and inverse result for the direct v
t for v = 0, 1, 2. Gadjiev and operators Ln using the test functions 1+t C ¸ akar [16] stated a Korovkin-type theorem for the uniform convergence of functions belonging to a suitable class Hω , where ω is the univariate function of modulus of continuity, by the sequence of Bleimann, Butzer and v Hahn t operators. In that paper, the authors used the test functions 1+t , where
v = 0, 1, 2. Altın et al. [8] introduced a tensor product bivariate extension of the Bleimann, Butzer and Hahn operators and studied approximation properties of these operators in the sense of Gadjiev and C ¸ akar [16]. Monotonicity properties of the sequence of the classical and q extension of the Bleimann, Butzer and Hahn operators were investigated, respectively, by Della Vecchia [14], and Do˘ gru and Gupta [15]. The operator Ln and some of its generalizations have been studied by several authors [1,3,5,6,11,18,21, 22]. Adell et al. [4] defined non-tensor product-type bivariate Bleimann, Butzer and Hahn operators as An (f, x, y) =
n−k−l n n−k n x k y
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