The Functional-Analytic Properties of the Limitq-Bernstein Operator

摘要

The limitq-Bernstein operatorBq,0<q<1, emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in theq-boson theory to describe the energy distribution in aq-analogue of the coherent state. Lately, the limitq-Bernstein operator has been widely under scrutiny, and it has been shown thatBqis a positive shape-preserving linear operator onC[0,1]with∥Bq∥=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 ofBqare studied. Our main result states that there exists an infinite-dimensional subspaceMofC[0,1]such that the restrictionBq|Mis an isomorphic embedding. Also we show that each such subspaceMcontains an isomorphic copy of the Banach spacec0.