On the Image of the Limit <i>q</I>-bernstein Operator

Abstract

The limit q-Bernstein operator B-q emerges naturally as an analogue to the Szasz-Mirakyan operator related to the Euler distribution. Alternatively, B-q comes out as a limit for a sequence of q-Bernstein polynomials in the case 0<q<1. Lately, different properties of the limit q-Bernstein operator and its iterates have been studied by a number of authors. In particular, it has been shown that B-q is a positive shape-preserving linear operator on C[0, 1] with parallel to B-q parallel to = 1, which possesses the following remarkable property: in general, it improves the analytic properties of a function. In this paper, new results on the properties of the image of B-q are presented. Copyright (C) 2009 John Wiley & Sons, Ltd.