On the injectivity with respect to q of the Lupaş q-transform



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Quaestiones Mathematicae


The Lupaş q-transform has first appeared in the study of the Lupaş q-analogue of the Bernstein operator. Given 0 < q < 1 and f ∈ C [0, 1], the Lupaş q-transform is defined by \begin{align*} \Lambda_q(f;x)=\prod_{k=0}^{\infty}\frac{1}{1+q^{k}x}\sum_{k=0}^{\infty} \frac{f(1-q^k)q^{k(k-1)/2}x^k}{(1-q)(1-q^2)\ldots(1-q^k)}, \quad x \geqslant 0. \end{align*} 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 Λq with respect to parameter q. More precisely, the conditions on f such that equality Λq (f ; x) = Λr (f ; x), x ⩾ 0 implies q = r have been established.


Published by Quaestiones Mathematicae;; Övgü Gürel Yılmaz, Recep Tayyip Erdogan University, Department of Mathematics, 53100, Rize, Turkey; Sofiya Ostrovska, Mehmet Turan, Atilim University, Department of Mathematics, Incek 06836, Ankara, Turkey.


Lupa¸s q-analogue of the Bernstein operators, Lupa¸s q-transform, analytic function, q-periodicity, 47B38, 30A99, 26C15