Direct and inverse theorems for Bernstein polynomials in the space of riemann integrable functions |
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| Authors: | Erich van Wickeren |
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| Institution: | 1. Lehrstuhl A für Mathematik, Aachen University of Technology, Templergraben 55, D-5100, Aachen, FRG
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| 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
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![$$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in 01]:|x - y \leqslant \delta \} ,}$$](/content/WN64174520K21570/365_2005_Article_BF01889606_TeX2GIFE1.gif)