Direct and inverse theorems for Bernstein polynomials in the space of riemann integrable functions

Abstract

Equivalence theorems concerning the convergence of the Bernstein polynomialsB n f are well known for continuous functionsf in the sup-norm. The purpose of this paper is to extend these results for functionsf, Riemann integrable on [0, 1], We have therefore to consider the seminorm $$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$ depending also on the increment δ>0. In terms of correspondingτ-moduli direct and inverse theorems are established, e.g., $$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$ Moreover, a similar result is obtained for the error functional appearing in the recently introduced definition of Riemann convergence.