A New Kantorovich Type q-analogue of the Balazs-Szabados Operators

This thesis consists of four chapters. In the first chapter, the introduction is offered. In the second chapter, we give the definitions, concepts and important theorems related with linear positive operators. We mention about the q-integers which are used to introduce q-analogue of the positive linear operators that have been intensive of research on approximation theory. After that, we mention about the definition of the operators which are introduced by Balázs and Szabados together. As well as, we shed light on various definitions of q-Balázs-Szabados operators, but we especially work on the new q-Balázs-Szabados operators which are defined by N. I. Mahmudov and denoted by (f x) n q ,  , . We calculate the formulas of (t x) m n q ,  , for m =1, 2, 3, 4 and we obtain the 1st, 2nd, 3rd and the 4th order moments of the new q-Balázs-Szabados operators. We also derive the recurrence formula of (t x) m n q ,  , in terms of         +  1 , , a x a x t n m n n q that represents a close connection between the new q-Balázs Szabados operators and the q-Bernstein operators. As well as, we estimate the 2nd order and the 4th order central moments of the operators (f x) n q ,  , , which have a great deal of importance of getting the results in approximation theory. Besides, we mention about the Kantorovich type q-analogue of the Balázs-Szabados operators (q-BSK operators) that have a nondecreasing restriction on f (x) to maintain the positivity property. In the third chapter, we construct a new Kantorovich type q-analogue of the Balázs-Szabados operators, n q, ( f x, )   . These new operators have an advantage compared to the previous ones, they maintain the positivity property without any restriction on f (x) . We give the recurrence formula for , ( , , 0 )   m n q t x m     and iv we calculate the formulas of (t x) m n q , ,   for m = 0,...,4 . Then, we give some significant auxiliary findings for the convergence properties of these operators n,q (f ; x)   . In terms of the usual modulus of continuous functions, we investigate the local approximation properties and we give Korovkin type approximation theorem for the operators n,q (f ; x).   We prove Voronoskaja type theorem and we present the convergence rate in terms of the usual Lipschitz functions, () LipM . In the fourth chapter, the conclusion is given. Keywords: q-calculus; q-Bernstein basis function; q-Bernstein operators; q-analogue of the Balázs-Szabados operators; moments; Voronovskaja theorem.