Preservation of Moduli of Continuity for Bernstein-Type Operators

Abstract

It is well known that many Bernstein-type operators preserve some properties of the functions on which they act, such as monotonicity, convexity, Lipschitz constants, etc. (cf. for instance [2]). In this paper, attention is focused on preservation of global smoothness, as measured by the usual moduli of continuity of first and second order. To the best of our knowledge, this problem has been studied by Kratz and Standtmüller in [11] for the first time. In this work the authors consider sequences (L _n ) _n≥1 of one-dimensional descrete operations satisfying certain moment assumptions and obtain estimates of the form

$$ \omega \left( {L_n f;h} \right) \leqslant c\omega \left( {f;h} \right), $$

((1))

where ω(f;.) stands for the usual first modulus of continuity of function f and c is a positive constant which depends on the particular family of operations considered, but not upon f nor n and h. They provide the estimate c ≤ 4 in some important examples, such as Bernstein, Szász and Baskakov operators.

Research supported by the University of the Basque Country and by the grant PB92-0437 of the Spanish DGICYT.