The Research About Q Analogue Of Szász-Beta-Stancu Operators And Modified Q-Baskakov Operators' Approximation Property

Operator approximation is an important branch of the approximation theory of functions,mainly study the convergence property and convergence rate of linear operator sequences(such as Bernstein operator,Baskakov operators,Szasz-Mirakjan operators,etc.),and describe the operator by its good properties of higher order derivatives,and study the approximation problem of some deformation operators'(such as Baskakov-Kantorovich Operators,etc.)approximation problem,saturation,and direct and inverse theorem.In the year of 2010,V.Gupta and Ali Aral defined a q-type Szász-Beta operator[1],it has many similar properties with q-Szász-Beta operator,and draws some classic conclusions,such as convergence theorem,approximation problems in weighted spaces and uniform approximation theorem.Besides,with the help of K-functional,its feature were described.In recent years,the study about stancu type operators,becomes popular in function approximation.Therefore we consider q analogue Szász-Beta-Stancu operator,to study various approximation problem of the operator,such as uniform convergence,approximation theorem.Baskakov operators is an important tool to study the function approximation issues.Thought of using the similar methods about study q analogue Szász-Beta-Stancu operator's approximation problem,we deeply studied a class of modified q-Baskakov operator's approximation properties in different function spaces,and reveal this kind of operator's monotonicity.The article is divided into the following sections: the first chapter mainly reviews the development history and current status of operator approximation,and some domestic and foreign research results about Szász operators and Baskakov operators.On the basis of research on the literatures about q type Szász-Beta operator approximation,in the second chapter,we draws a new operator by making appropriate amendments on the q-Szász-Beta operator,namely q analogue of Szász-Beta-Stancu operator,and estimates its moments,given its Vorontsovskaya type theorem.In the third chapter,with the help of central moments and modulus of smoothness,and other tools,we obtained a positive approximation theorem,pointwise convergence rate and weighted approximation properties on the q analogue of Szász-Beta-Stancu operator.The fourth chapter studies modified q-Baskakov operators,by computing the central moments of the operator,we obtain its monotonous property,using a continuous modules and other tools,we obtained their approximation theorems for different functions like Lipschitz functions,making it suitable for a wider of function space,expanding the scope of the study.The fifth chapter summarizes the issues raised in the main text,explounds the purpose and significance,and make a predict about whether itcan continue to improve the previous results,and prospect the possible future research directions.