The Functional-Analytic Properties of the Limit <i>q</I>-bernstein Operator

Abstract

The limit q-Bernstein operator B-q, 0 < q < 1, emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. Lately, the limit q-Bernstein operator has been widely under scrutiny, and 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. Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties of B-q are studied. Our main result states that there exists an infinite-dimensional subspace M of C[0, 1] such that the restriction B-q vertical bar(M) is an isomorphic embedding. Also we show that each such subspace M contains an isomorphic copy of the Banach space c(0).