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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

(L n f)(x)= k=0 n p n,k (x)λ n,k (f),

fC[0,1], such that L n preserves linear functions: L n (t-x,x)=0 and L n ((t-x) 2 ,x)=n -1 φ 2 (x), x[0,1], where φ 2 is not necessarily a concave function on [0,1]. 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:
41A25Rate of convergence, degree of approximation