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Approximation by iterative combination of exponential type operators. (English) Zbl 0605.41023

The iterative combination \(S_{\lambda,k,m}\) of the exponential type operator \(S_{\lambda}\) is defined by \[ S_{\lambda,k,m}(f,t)=\sum^{k}_{r=1}\frac{(- 1)^{r+1}}{m^{\beta}(m,r)}\left( \begin{matrix} k+m\\ k-r\end{matrix} \right)S_{\lambda}^{r+m}(f,t), \] where \(k\in {\mathbb{N}}\), \(m\in I_ 0\) (non-negative integers), \(\lambda \in {\mathbb{R}}_+\) (positive real numbers) and \(\beta\) (m,r) is the \(\beta\)-function. The authors prove the basic convergence theorem and an asymptotic formula for \(S_{\lambda,k,m}(f,t)\). The last result is proved using the properties of another sequence of linear approximating functions, viz, Steklov means.

MSC:

41A36 Approximation by positive operators
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