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On certain linear positive operators in exponential weighted spaces. (English) Zbl 1111.41017
Let $$C_q$$, $$q>0$$, be the space associated with the weighted function $v_q(x):=e^{-qx},\;\;x\in R_0=[0,+\infty )$ which consists of all real-valued functions $$f$$ continuous on $$R_0$$ for which $$v_qf$$ is uniformly continuous and bounded on $$R_0$$. The norm on $$C_q$$ is defined by $\| f\| _q\equiv \| f(\cdot )\| _q:=\sup _{x\in R_0}v_q(x)| f(x)| .$ For $$r\in N$$ and $$q>0$$, the author defines the following class of modified Szász-Mirakyan operators in the space $$C_{2q}$$: $\left(A_n^{(q,r)}f\right)(x):={1\over g((nx+1)^2;r)}\sum _{k=0}^{\infty }{(nx+1)^{2k}\over(k+r)!}f\left({k+r\over n(nx+1)+2q}\right)$ for $$x\in R_0$$, $$n\in \mathbb N$$, where $$g(t;r)=\sum _{k=0}^{\infty }{t^k\over (k+r)!}$$, $$t\in R_0$$. He proves that:
a)
these operators are linear and positive on $$C_{2q}$$
b)
if one fixes $$q>0$$ and $$r\in \mathbb N$$, there exists a positive constant $$M_1$$ such that for every $$f\in C_{2q}^1:=\{f\in C_{2q}: f'\in C_{2q}\}$$, $\| A_n^{(q,r)}f-f\| _{2q}\leq {M_1\over n}\| f'\| _{2q},\;\;n\in \mathbb N,$
c)
if one fixes $$q>0$$ and $$r\in N$$, there exists a positive constant $$M_2$$ such that for every $$f\in C_{2q}$$, $\| A_n^{(q,r)}f-f\| _{2q}\leq M_2\omega _1(f;C_{2q};1/n),\;\;n\in \mathbb N,$ where $$\omega _1(f;C_{2q};t):=\sup _{0\leq h\leq t}\| \triangle _hf(\cdot )\| _{2q}$$.

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