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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
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