On The Approximation Of Two Types Of Q-Szasz-Mirakyan Operators On Two Variables

In 1950, Otto Szasz [1] generalized the Bernstein operators to infinite interval, which is calledSzasz-Mirakyan operators. In 1998, Lucyna. R and Mariola S proposed the Szasz-Mirakyanoperator on two variables in the paper[3]. In 2006, the authors Ali Aral. and Vijoy Gupta developedthe Szasz-Mirakyan operator on q-integers, and studied the approximation property of the operator,such as the Korovikins type theorems and Voronovskaya theorems and continued to study theq-derivatives[5,7]. Later, N. I Mahmudov[52,53] proposed another type of q-Szasz-Mirakyanoperators and the q-Kantorovich Operators.In the dissertation, we generalized two types of q-Szasz-Mirakyan operators, which are calledpartial q-Szasz-Mirakyan operaors and study the Voronovskaya theorems of the operators andcharacterize the approximation property of one of them by the weighted smoothness. In the firstchapter, the dissertation makes a brief introduction to the background and primaryresults of operator approximation and q-analysis. In the secondly chapter, the authorproposes the q-Szasz-Mirakyan operators on functions of two variables and studied theapproximation property of the operators. The Voronovskaya type theorem is proved. However, thereis some limitation of the operator, because the domains of the functions under the operators areindependent of thebn andc nin the definitions. The author proposes the q-Szasz operator on twovariables in chapter 3 to avoid the limitation. The similar properties with q-Szasz-Mirakyanoperators on two variables can be found on the operator. In Chapter 4, the author redefines a newkind of partial q-derivatives, and proposes a problem that has not been resolved by the paper.