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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 $$\| f\| _p=\| f(\cdot)\| _p:=\sup_{x\in \mathbb R_0}w_p(x)| f(x)|$$.

##### MSC:
 41A36 Approximation by positive operators