A Survey of Results on the Limit -Bernstein Operator

Sofiya Ostrovska

doi:10.1155/2013/159720

Abstract

The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

Highlights

The limit q-Bernstein operator comes out as an analogue of the Szasz-Mirakyan operator related to the Euler probability distribution, called the q-deformed Poisson distribution
The latter is used in the q-boson theory, which is a q-deformation of the quantum harmonic oscillator formalism [4]
The q-analogue of the boson operator calculus has proved to be a powerful tool in theoretical physics, providing explicit expressions for the representations of the quantum group SUq(2), which itself is known to play a profound role in a variety of different problems

Summary

Introduction

The limit q-Bernstein operator comes out as an analogue of the Szasz-Mirakyan operator related to the Euler probability distribution, called the q-deformed Poisson distribution (see [1,2,3]). It can be readily seen from the definition that Bq possesses the end-point interpolation property: Bq (f; 0) = f (0) , Bqf (1) = f (1) It is commonly known in the field that Bq leaves invariant linear functions and maps a polynomial of degree m to a polynomial of degree m (see Theorem 26). Additional properties of this operator will be considered in the present paper. The q-Bernstein polynomials possess the end-point interpolation property, leave invariant linear functions, admit representation with the help of q-differences, and are degree-reducing on polynomials.

Probabilistic Approach

Approximation Properties of Bq

Functional-Analytic Properties of the Limit q-Bernstein Operator

The Improvement of Analytic Properties under the Limit q-Bernstein Operator

Concluding Remarks