Rates of ideal convergence for approximation operators
This paper presents some convergence results on the Korovkin-type approximation theory when the notion of \({\mathcal I}\)-convergence is used, where \({\mathcal I}\) is an ideal in \(\mathbb N\) such that \(\{n\}\in{\mathcal I}\), \(n\in\mathbb N\). This new type of convergence was introduced by \textit{P. Kostyrko, T. Šalát} and \textit{W. Wilczynski} in [Real Anal. Exch. 26, No.2, 669-685 (2001; Zbl 1021.40001)], and it generalizes both statistical and \(A\)-statistical convergence. In particular, the ideal convergence reduces to the ordinary convergence when the ideal \({\mathcal I}\) is the class of all finite subsets of \(\mathbb N\). Let \({\mathcal U}^r=U_1\times \cdots\times U_r\), where every \(U_i\) is an arbitrary interval of \(\mathbb R\). The authors consider the space \(C_g({\mathcal U}^r)\) of continuous functions \(f\) on \({\mathcal U}^r\) with \(|f(\mathbf{x})|\leq M g(\mathbf{x})\), for some \(M>0\) and all \(\mathbf{x}\in {\mathcal U}^r\), where \(g\) is a given continuous non-negative function on \(\mathbb R^r\), such that \(g(\mathbf{x})\geq 1\), for all \(\mathbf{x}\in\mathbb R^r\). This space is endowed with the norm \(\|f\|_g=\sup_{\mathbf{x}\in{\mathcal U}^r}|f(\mathbf{x})|/g(\mathbf{x})\). Some results of Korovkin-type are proved for operators of the form \(L_n(f;\mathbf{x})=\int_{{\mathcal U}^r}f(\mathbf{y})\,\mathbf{d}\mu_{n,\mathbf{x}}(\mathbf{y})\), where \(\mu_{n,\mathbf{x}}\) is a collection of product measures on \({\mathcal U}^r\). The approximation is given in the sense of the \({\mathcal I}\)-convergence, and the test functions are \(e_0(\mathbf{x})=1\), \(e_i(\mathbf{x})=x_i\), \(i=1,\dots,r\), \(e_{r+1}(\mathbf{x})=x_1^2+\cdots+x_r^2\).
scientific article; zbMATH DE number 5728331
scientific article