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Article On the Lupas q-transform of Unbounded Functions(Walter de Gruyter Gmbh, 2023) Ostrovska, Sofiya; Turan, MehmetThe Lupa , s q-transform comes out naturally in the study of the Lupa , s q-analogue of the Bernstein operator. It is closely related to the Heine q-distribution which has a numerous application in q-boson operator calculus and to the Valiron method of summation for divergent series. In this paper, the Lupa , s q-transform (lambda(q)f)(z), q is an element of (0, 1), of unbounded functions is considered in distinction to the previous researches, where only the case f is an element of C[0, 1] have been investigated. First, the condition for a function to possess the Lupa , s q-transform is presented. Also, results concerning the connection between growth rate of the function f (t) as t -> 1(-) and the growth of its Lupa , s q-transform (lambda(q)f)(z) as z -> infinity are established. (c) 2023 Mathematical Institute Slovak Academy of SciencesArticle Citation - WoS: 5Citation - Scopus: 5Analytical Properties of the Lupas q-transform(Academic Press inc Elsevier Science, 2012) Ostrovska, SofiyaThe Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Given q is an element of (0, 1), f is an element of C left perpendicular0, 1right perpendicular, the q-Lupas transform off is defined by (Lambda(q)f) (z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k -1)/2)/(q; q)(k)z(k). The transform is closely related to both the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus, and to Valiron's method of summation for divergent series. In general, Lambda(q)f is a meromorphic function whose poles are contained in the set J(q) := {-q(-j)}(j=0)(infinity). In this paper, we study the connection between the behaviour of f on leftperpendicular0, 1right perpendicular and the decay of Lambda(q)f as z -> infinity. (C) 2012 Elsevier Inc. All rights reserved.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: 1Citation - Scopus: 1Geometric Properties of the Lupas q-transform(Tusi Mathematical Research Group, 2014) Ostrovska, SofiyaThe Lupas q-transform emerges in the study of the limit q-Lupas operator. This transform is closely connected to the theory of positive linear operators of approximation theory, the q-boson operator calculus, the methods of summation of divergent series, and other areas. Given q is an element of (0, 1), f is an element of C[0, 1], the Lupas q-transform of f is defined by: [GRAPHICS] where [GRAPHICS] The analytical and approximation properties of A(q) have already been examined. In this paper, some properties of the Lupas q-transform related to continuous linear operators in normed linear spaces are investigated.Article On the Image of the Lupas q-analogue of the Bernstein Operators(Springernature, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThe Lupas q-analogue, R-n,R-q, is historically the first known q-version of the Bernstein operator. It has been studied extensively in different aspects by a number of authors during the last decades. In this work, the following issues related to the image of the Lupas q-analogue are discussed: A new explicit formula for the moments has been derived, independence of the image R-n,R-q from the parameter q has been examined, the diagonalizability of operator R-n,R-q has been proved, and the fact that R-n,R-q does not preserve modulus of continuity has been established.
