The Functional-Analytic Properties of the Limit q-Bernstein Operator

Abstract EN

The limit q-Bernstein operator Bq, 0

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 Bq is a positive shape-preserving linear operator on C[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 of Bq are studied.

Our main result states that there exists an infinite-dimensional subspace M of C[0,1] such that the restriction Bq|M is an isomorphic embedding.

Also we show that each such subspace M contains an isomorphic copy of the Banach space c0.