The Functional-Analytic Properties of the Limit q-Bernstein Operator
The limit q-Bernstein operator Bq, 0<q<1, emerges naturally as a modification of the Szász-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 oper...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2012-01-01
|
| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/280314 |
| Summary: | The limit q-Bernstein operator Bq, 0<q<1, emerges naturally as a modification of the Szász-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 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. |
|---|---|
| ISSN: | 0972-6802 1758-4965 |