On modified Baskakov operators on weighted space (Q2770952)

Given \(A>0\) and a sequence of analytic functions \(\{\varphi_n:[0,A]\rightarrow \mathbb{R}\}_{n\in \mathbb{N}}\), the well known generalized Baskakov operators are defined for any \(n\in \mathbb{N}\), \(x\in[0,A)\) and \(f:[0,\infty)\rightarrow R\) by NEWLINE\[NEWLINE L_nf(x)=\sum_{k=0}^\infty f\left({k\over n}\right)\varphi_n^{(k)}(x) {(-x)^k\over k!}. NEWLINE\]NEWLINE In this paper the author presents a new modification of the Baskakov operators by considering each function \(\varphi_n\) defined on \([0,b_n]\) for a sequence \(\{b_n\}_{n\in \mathbb{N}}\subseteq [0,\infty)\) with finite or infinite limit (instead of the same domain, \([0,A]\), for all of them), under the assumption that they satisfy the classical restrictions (namely, \(\varphi_n(0)=1\) and \((-1)^k\varphi_n^{(k)}(x)\geq 0\) for all \(n\), \(k\) and \(x\)) together with the following further condition: there exists a positive integer \(m\) such that \(\varphi_n^{(k)}(x)=-n\varphi_{n+m}^{(k-1)}(x)(1+\alpha_{k,n}(x))\), where \(\alpha_{k,n}(0)=O({1\over n^k})\) and \({\varphi_n^{(k)}(0) \over n^k}=(-1)^k+O({1\over n})\), for all \(n\), \(k\) and \(x\). Given \(\rho(x)=1+x^2\), consider the space \(C_\rho^k\) that consists of all continuous functions on \([0,\infty)\) such that \(\lim_{x\rightarrow \infty}{f(x)\over {\rho(x)}}=k\in \mathbb{R}\), equipped with the norms \(\|f\|_{\rho^s}=\sup_{x\in[0,\infty)} {{|f(x)|}\over{\rho^s(x)}}\), \(s\in \mathbb{N}\). The convergence for functions of \(C_\rho^k\) with respect to the weighted norms \(\|\cdot\|_{\rho^s}\) is studied. More precisely, the author affirms that for \(f\in C_\rho^k\) we have \(\|L_nf-f\|_\rho\rightarrow 0\) and \(\|L_nf-f\|_{\rho^3}\leq K\Omega_n(f,n^{-{1\over 4}})\) (here we consider a function defined on \([0,b_n]\) to vanish out of its domain), where \(\Omega_n(f,\delta)=\sup_{|h|\leq \delta,x\in[0,b_n]} {{|f(x+h)-f(x)|}\over {(1+h^2)(1+x^2)}}\). Besides, if \(f^{(r-1)}\in C_\rho^k\) and \(f^{(r)}\in \text{ Lip}_\alpha[0,\infty)\) (where \(\text{ Lip}_\alpha[0,\infty)\) stands for the classical space of Hölder continuous functions of order \(\alpha\in(0,1]\)) the author asserts that \(\sup_{x\in[0,b_n]}{{|L_n^{(r)}f(x)-f^{(r)}(x)|}\over {1+x^\alpha}}=O({1\over n})\). The paper contains important mistakes and the notation and all enunciates of results and proofs should be carefully revised.