The Distance Between Two Limit <i>q</I>-bernstein Operators

Abstract

For q is an element of (0, 1), let B-q denote the limit q-Bernstein operator. The distance between B-q and B-r for distinct q and r in the operator norm on C[0, 1] is estimated, and it is proved that 1 <= parallel to B-q - B-r parallel to <= 2, where both of the equalities can be attained. Furthermore, the distance depends on whether or not r and q are rational powers of each other. For example, if r(j) not equal q(m) for all j, m is an element of N, then parallel to B-q - B-r parallel to = 2, and if r = q(m) for some m is an element of N, then parallel to B-q - B-r parallel to = 2(m - 1)/m.