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Article On the Injectivity With Respect To q of the Lupas q-transform(Taylor & Francis Ltd, 2024) Yilmaz, Ovgue Gurel; Ostrovska, Sofiya; Turan, MehmetThe Lupas q-transform has first appeared in the study of the Lupas q-analogue of the Bernstein operator. Given 0 < q < 1 and f is an element of C[0, 1], the Lupas q-transform is defined by Lambda(q)(f; x) Pi(infinity)(k=0) 1/1 + q(k)x Sigma(k=0)f(1 - q(k))q(k(k-1)/2)x(k)/(1 - q)(1 - q(2)) center dot center dot center dot (1 - q(k)), x >= 0. During the last decades, this transform has been investigated from a variety of angles, including its analytical, geometric features, and properties of its block functions along with their sums. As opposed to the available studies dealing with a fixed value of q, the present work is focused on the injectivity of Lambda(q) with respect to parameter q. More precisely, the conditions on f such that equality Lambda(q)(f; x) = Lambda(r)(f; x); x >= 0 implies q = r have been established.Article 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].
