On the Asymptotic Approximation with Bivariate Operators of Bleimann, Butzer, and Hahn

Abstract

The concern of this paper is a recent generalization L n ( f (t 1 , t 2 ); x, y) for the operators of Bleimann, Butzer, and Hahn in two variables which is distinct from a tensor product. We present the complete asymptotic expansion for the operators L n as n tends to infinity. The result is in a form convenient for applications. All coefficients of n &k (k=1, 2, ...) are calculated explicitly in terms of Stirling numbers of the first and second kind. As a special case we obtain a Voronovskaja-type theorem for the operators L n .

Asymptotic formulae for bivariate Mellin convolution operators

“…Analogously we have (1,1) and so finally the assertion follows. Now we will discuss the case when f ∈ L p (μ), 1 ≤ p < +∞.…”

“…[4] and for bivariate case see [1], [2], [6], [13]). In particular we mention the paper [18] in which asymptotic expansions for certain one dimensional convolution operators are given.…”

An answer to Hermann's conjecture on Bleimann–Butzer–Hahn operators

“…Their properties were studied in many papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. As was already shown in [6] the operators L n approximate bounded continuous functions on [0, ∞).…”

Localization results for generalized Baskakov/Mastroianni and composite operators

“…(C2) (Mastroianni operators) For every n ∈ N and k ∈ N 0 there exist p(n, k) ∈ N and α n,k : [0, ∞) → R such that D i+k φ n (x) = (−1) k α n,k (x)D i φ p(n,k) (x) (2) for every i ∈ N 0 and x ∈ [0, ∞) in such a way that lim n→∞ n p(n,k) = lim n→∞ α n,k (x)…”