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On simultaneous approximation by Szász-beta operators. (English) Zbl 0846.41022

The authors have defined a new sequence of linear positive operators by coupling Szász and beta operators as

${B}_{n}\left(f;x\right)=\sum _{k=0}^{\infty }{p}_{n,k}\left(x\right){\int }_{0}^{\infty }{b}_{n,k}\left(y\right)f\left(y\right)dy,\phantom{\rule{2.em}{0ex}}x\in \left[0,\infty \right)$

where

${p}_{n,k}\left(x\right)={e}^{-nx}\frac{{\left(nx\right)}^{k}}{k!}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{b}_{n,k}\left(y\right)=\frac{1}{\beta \left(k+1,n\right)}\frac{{y}^{k}}{{\left(1+y\right)}^{n+k+1}}·$

The main result of the paper is the Woronowskaja type asymptotic formula in simultaneous approximation for Lebesgue integrable functions. The next result is the local estimate for the $r$th derivative of the function. Finally, the claim that the main theorem has improved the earlier results on modified Baskakov and Szász operators by H. S. Kasana, P. N. Agrawal and V. Gupta [Approximation Theory Appl. 7, No. 2, 65-82 (1991; Zbl 0755.41024)].

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