On the Injectivity With Respect To <i>q</I> of the Lupas <i>q</I>-transform

Abstract

The 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.