Steckin-marchaud-type inequalities in connection with bernstein polynomials

Abstract

The purpose of this paper is to derive the estimate (0≤α≤2,n∈N,ϕ(x)=[x(1−x)]^1/2)

$$\omega _\alpha (n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,f) \leqslant M_\alpha n^{ - 1} \sum\limits_{k = 1}^n {\left\| {\varphi ^{ - \alpha } (B_k f - f)} \right\|} c$$

in terms of the modulus of continuity (of second order)

$$\omega _\alpha (t,f): = \sup \{ \varphi ^{ - \alpha } (x)|\Delta _{h\varphi (x)}^ * f(x)|:x,x \pm h\varphi (x) \in [0,1],0< h \leqslant t\} $$

and the Bernstein polynomial B_nf for ϕ^−αf∈C[0,1].