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Article Citation - WoS: 2Citation - Scopus: 2The Impact of the Limit q-durrmeyer Operator on Continuous Functions(Springer Heidelberg, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThe 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.Article On the Eigenstructure of the Modified Bernstein Operators(Taylor & Francis inc, 2022) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetStarting 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.
