2 results
Search Results
Now showing 1 - 2 of 2
Article Citation - WoS: 1Citation - Scopus: 2Functions Whose Smoothness Is Not Improved Under the Limit q-bernstein Operator(Springer, 2012) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler probability distribution. At the same time, this operator serves as the limit for a sequence of the q-Bernstein polynomials with 0 < q < 1. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. Its approximation, spectral, and functional-analytic properties, probabilistic interpretation, the behavior of iterates, and the impact on the analytic characteristics of functions have been examined. It has been proved that under a certain regularity condition, B-q improves the smoothness of a function which does not satisfy the Holder condition. The purpose of this paper is to exhibit 'exceptional' functions whose smoothness is not improved under the limit q-Bernstein operator. MSC: 26A15; 26A16; 41A36Article Citation - WoS: 3Citation - Scopus: 4The Unicity Theorems for the Limit Q-Bernstein Operator(Taylor & Francis Ltd, 2009) Ostrovska, SofiyaThe limit q-Bernstein operator [image omitted] emerges naturally as a q-version 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. The limit q-Bernstein operator has been widely studied lately. It has been shown that [image omitted] is a positive shape-preserving linear operator on [image omitted] with [image omitted] Its approximation properties, probabilistic interpretation, the behaviour of iterates, eigenstructure and the impact on the smoothness of a function have been examined. In this article, we prove the following unicity theorem for operator: if f is analytic on [0, 1] and [image omitted] for [image omitted] then f is a linear function. The result is sharp in the following sense: for any proper closed subset [image omitted] of [0, 1] satisfying [image omitted] there exists a non-linear infinitely differentiable function f so that [image omitted] for all [image omitted].
