Direct and inverse theorems for Bernstein polynomials in the space of Riemann integrable functions (Q581792)

For \(f\in C[0,1]\) let \(B_ n\), \(n\in {\mathbb{N}}\), denote the nth Bernstein operator and \(\omega (f,\delta)_{\infty}\) the second weighted Ditzian- Ivanov-Totik modulus of continuity with respect to the sup-norm. The objective of this paper is to extend the well-known equivalence theorem \[ (*)\quad \omega (f,\delta)_{\infty}=O(\delta^{\nu}),\quad 0<\nu <2\quad \Leftrightarrow \quad \| B_ nf-f\|_{\infty}=O(n^{- \alpha /2}) \] to the larger space \(R=R[0,1]\) of Riemann integrable functions on [0,1]. This turned out to be possible e.g. if in (*) the sup-norm is replaced by a certain family of seminorms depending on the increment \(\delta >0\) and if the modulus of continuity is replaced by a suitably modified \(\tau\)-modulus in the sense of Ivanov. A second similar equivalence holds for weighted norms and a corresponding I-modulus. Finally Theorem 4.2 gives a pair of direct and converse results for the Riemann error of \(B_ nf-f\) on R.