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Direct estimates and \(L_p\)-approximation properties for Agratini operators and their integral form. (English) Zbl 1073.41018

Let \(\{\varphi_n\}\) be a sequence of analytic functions in a domain \(\Omega\) which contains the closed disk of radius \(a\), where \(a\in(0,1)\). The authors assume that \(\varphi_n(0)>0\), and that \(\varphi^{(k)}_n(0)=\alpha_n(k+n+\beta_k)(1+\gamma_{n,k}) \varphi^{(k-1)}_n(0)\), \(k\geq1\), where \(1\leq\alpha_n=1+O(1/n)\), \(0\leq\beta_k\leq\beta_{k+1}+1\), and \(0\leq\gamma_{n,k}=O(1/n)\). The authors also assume that \(\varphi^{(k)}_n(0)\geq0\), but this trivially follows from the previous assumptions, in fact it follows that \(\varphi^{(k)}_n(0)>0\). Given \(f\in C[0,a]\), the authors denote \[ D_n(f,x):=\frac1{\varphi_n(x)}\sum_{k=0}^\infty\varphi^{(k)}_n(0) \frac{x^k}{k!}f\biggl(\frac k{k+n+\beta_k}\biggr),\quad n\geq1, \] and give some estimates on the degree of approximation of \(f\) in the sup-norm on \([0,a]\), by these positive operators. Unfortunately, the reviewer does not understand how the operators are defined if \(a<1\), since the argument \(\frac k{k+n+\beta_k}\to1\) as \(k\to\infty\).
The authors also consider the Kantorovich variant of the operators in order to obtain the degree of approximation in the \(L_p\)-norm of integrable functions on \([0,1]\). Here the definition is valid, but the operators are defined only for \(x\in[0,a]\), so it is not clear how the authors claim these are bounded operators from \(L_p[0,1]\) to \(L_p[0,1]\).

MSC:

41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
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