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On the approximation of unbounded functions by Bernstein polynomials. (English) Zbl 1111.41010

Summary: We show the existence of an operator of the type

$\left({L}_{n}f\right)\left(x\right)=\sum _{k=0}^{n}{p}_{n,k}\left(x\right){\lambda }_{n,k}\left(f\right),$

$f\in C\left[0,1\right]$, such that ${L}_{n}$ preserves linear functions: ${L}_{n}\left(t-x,x\right)=0$ and ${L}_{n}\left({\left(t-x\right)}^{2},x\right)={n}^{-1}{\varphi }^{2}\left(x\right)$, $x\in \left[0,1\right]$, where ${\varphi }^{2}$ is not necessarily a concave function on $\left[0,1\right]$. The generalized Bernstein-type operators ${L}_{n}$ have been introduced and studied in [I. Gavrea and D. H. Mache, Generalization of Bernstein-type approximation methods. Müller, Manfred W. (ed.) et al., Approximation theory. Proceedings of the 1st international Dortmund meeting IDoMAT 95 held in Witten, Germany, March 13–17, 1995. Berlin: Akademie Verlag. Math. Res. 86, 115–126 (1995; Zbl 1005.41500)] and [M. Felten, J. Approximation Theory 94, No. 3, 396–419 (1998; Zbl 0913.41008)], respectively. In this paper we establish direct and converse theorems for the above-mentioned operators under more general conditions concerning the weight functions.

##### MSC:
 41A25 Rate of convergence, degree of approximation