Browsing by Author "Yilmaz, Ovgu Gurel"

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Citation - WoS: 2
Citation - Scopus: 2
The Impact of the Limit q-durrmeyer Operator on Continuous Functions
(Springer Heidelberg, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, Mehmet; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
The limit q-Durrmeyer operator, D-infinity,D-q, was introduced and its approximation properties were investigated by Gupta (Appl. Math. Comput. 197(1):172-178, 2008) during a study of q-analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of D-infinity,D-q. The interrelation between the analytic properties of a function f and the rate of growth for D(infinity,q)f are established, and the sharpness of the obtained results are demonstrated.
On the Continuity in q of the Family of the Limit q-durrmeyer Operators
(de Gruyter Poland Sp Z O O, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, Mehmet; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
This study deals with the one-parameter family {D-q}(q is an element of[0,1]) of Bernstein-type operators introduced by Gupta and called the limit q-Durrmeyer operators. The continuity of this family with respect to the parameter q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that {D-q}(q is an element of[0,1]) is continuous in the strong operator topology for all q is an element of [0, 1]. When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 and fails at all q is an element of (0, 1]. In addition, a few estimates for the distance between two limit q-Durrmeyer operators have been derived in the operator norm on C[0, 1].
On the Eigenstructure of the Modified Bernstein Operators
(Taylor & Francis inc, 2022) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, Mehmet; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
Starting from the well-known work of Cooper and Waldron published in 2000, the eigenstructure of various Bernstein-type operators has been investigated by many researchers. In this work, the eigenvalues and eigenvectors of the modified Bernstein operators Q(n) have been studied. These operators were introduced by S. N. Bernstein himself, in 1932, for the purpose of accelerating the approximation rate for smooth functions. Here, the explicit formulae for the eigenvalues and corresponding eigenpolynomials together with their limiting behavior are established. The results show that although some outcomes are similar to those for the Bernstein operators, there are essentially different ones as well.
On the Image of the Lupas q-analogue of the Bernstein Operators
(Springernature, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, Mehmet; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
The 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.
Shape-Preserving Properties of the Limit q-durrmeyer Operator
(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, Mehmet; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
The present work aims to establish the shape-preserving properties of the limit q- Durrmeyer operator, D q for 0 < q < 1. It has been proved that the operator is monotonicity- and convexity-preserving. What is more, it maps a function m - convex along {q (j)}(infinity)(j =0) to a function m - convex along any sequence { xq( j )}(infinity)(j =0) , x is an element of (0, 1). (c) 2024 Elsevier Inc. All rights reserved.