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Approximation by some linear positive operators in polynomial weighted spaces. (English) Zbl 1079.41023
It is well-known the following Szász-Mirakyan operator: \(S_n(f,x):=e^{-nx}\sum^\infty_{k=0}\frac{(nx)^k}{k!} f(\frac k n)\), \(f\in C_p\), \(x\in \mathbb R_0:=[0,\infty)\), \(n\in \mathbb N:=\{1,2,\dots\}\), where \(C_p\) with some fixed \(p\in \mathbb N_0:=\{0,1,2,\dots\}\) is the set of all real-valued functions \(f\), continuous on \(\mathbb R_0\) and such that \(w_p f\) is uniformly continuous and bounded on \(\mathbb R_0\); the weight function \(w_p\) is defined by \(w_0(x):=1\) and \(w_p(x):=(1+x^p)^{-1}\) if \(p\geq 1\).
The actual construction of the operator \(S_n\) and its various modifications requires estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. Therefore the author proposes the introduction of the following operator: \[ L_n(f,x):=\frac 1{(1+(x+n^{-1})^2)^n}\sum^n_{k=0}\binom{n}{k}(x+n^{-1})^{2k}f\left(\frac kn\cdot \frac{1+(x+n^{-1})^2}{x+n^{-1}}\right), \] where \(f\in C_p\), \(x\in \mathbb R_0\), \(n\in \mathbb N\).
The author proves – among others – that \(L_n\) gives better degree of approximation of functions \(f\in C_p\) than the usual Szász-Mirakyan operator \(S_n\). The central result of the paper is the following Theorem. Let \(p\in \mathbb N_0\) be a fixed number. Then there exists \(M(p)>0\) such that for every \(f\in C_p\) and \(n\in \mathbb N\) we have \[ \| L_n(f,\cdot)-f(\cdot)\| _p\leq M(p)\omega_1\left(f;C_p;\frac 1{\sqrt n}\right), \] where \(\omega_1(f;C_p;t)=\sup_{0<h\leq t}\| f(\cdot +h)-f(\cdot)\| _p\), \(t\in \mathbb R_0\) and \(\| f\| _p=\| f(\cdot)\| _p:=\sup_{x\in \mathbb R_0}w_p(x)| f(x)| \).

41A36 Approximation by positive operators