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On certain GBS-Durrmeyer operators based on q -integers

In the present paper we introduce the GBS (Generalized Boolean Sum) operators of Durrmeyer type based on q -integers and the approximation of B-continuous functions using the above operators is studied. In addition, a uniform convergence theorem is established and the degree of approximation in terms of mixed modulus of continuity is evaluated. | Turk J Math (2017) 41: 368 – 380 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1601-34 Research Article On certain GBS-Durrmeyer operators based on q -integers 1 ˘ Dan BARBOSU , Ana-Maria ACU2 , Carmen Violeta MURARU3,∗ Department of Mathematics and Informatics, North University Center at Baia Mare, Technical University of Cluj-Napoca, Baia Mare, Romania 2 Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Sibiu, Romania 3 Department of Mathematics, Informatics and Education Sciences, ”Vasile Alecsandri” University of Bac˘ au, Bac˘ au, Romania 1 Received: 10.01.2016 • Accepted/Published Online: 12.05.2016 • Final Version: 03.04.2017 Abstract: In the present paper we introduce the GBS (Generalized Boolean Sum) operators of Durrmeyer type based on q -integers and the approximation of B-continuous functions using the above operators is studied. In addition, a uniform convergence theorem is established and the degree of approximation in terms of mixed modulus of continuity is evaluated. The study contains in the last section numerical considerations regarding the constructed operators based on MATLAB algorithms. Key words: Positive linear operator, Durrmeyer-type operators, GBS operator, B -continuous function, mixed modulus of continuity, q -integers 1. Introduction In [12, 13], Karl B¨ogel introduced the concepts of B-continuous and B-differentiable functions. The GBS (Generalized Boolean Sum) operators are used in the uniform approximation of B -continuous functions. The term GBS operators was introduced by Badea et al. in [6]. In recent years, several researchers have made significant contributions in this area of approximation theory [1, 3, 11, 18, 21-24, 27, 29]. In this paper we study the uniform approximation of B -continuous functions using GBS operators of Durrmeyer type based on q -integers. The notions like GBS operators, B -continuous functions, and mixed modulus of