Research On The Approximation Properties Of Szász-Bézier Operators

The practical significance of function approximation theory becoming more and more extensive,such as the needs of data processing,weather forecast trend chart,image signal analysis,curve and surface design,we will further analyze and study the approximation problem.Considering the different forms and different constraints of complex function space,we will use simple operators as approximation tools to approximate the function-s to be studied,estimate the approximation error,find the best approximator and its characteristics,so as to simplify the research work.At present,we will focus on an im-portant branch of this field:operator approximation theory.The research idea of this theory is:using traditional classical operators(such as Baskakov operators,Bernstein operators,Szász operators,etc)and their modified types to approximate complex func-tions such as continuous functions,measurable functions,bounded variation functions and Lip functions.In this paper,taking the Szász operator as an example,some different shape parameters are added to improve the definition form of the operator.Starting from three types of Bézier operators(including ?-type,?-type,??-type)and using the e-quivalence relationship between modulus of smoothness and K-functional,the recurrence formula,the Bernstein inequality,the Cauchy-Schwarz inequality and the Holder inequal-ity for analysis.Firstly,we study the approximation direct theorem,voronovskaja-type weak inverse theorem of A-Szász-Mirakian operators and A-Szász-Kantorovich operators in CB[0,?),and the approximation direct theorem of these two kinds of operators to Lip function class.Secondly,we study the direct,inverse and equivalence theorems of?,?-Szász type operators in CB[0,?)space.Finally,based on the direct theorems of?,?-Bernstein operators,we will improve the definition of the Bernstein-Bézier opera-tors,and obtain the converse and equivalence theorems of the operators in C[0,1].This paper is divided into the following four chapters:In chapter 1,we give the basic concepts that will be used in the paper such as function of space,symbols,norm,modulus of smoothness and K-functional,and the definitions of Szász operator,Bernstein operator and their modified types.In chapter 2,the moment estimates of A-Szász type operator are obtained by cal-culation,and the approximation theorem of the operators in CB[0,?)space is obtained by combining the relationship between K-functional and modulus of smoothness,and the direct theorem of A-Szász type operators in Lip function space is obtained by using the research method of Bernstein type operator.In chapter 3,we first study the important properties of ??-Szász type operators,and then use the Cauchy-Schwarz inequality and so on to obtain the direct,inverse and equivalence theorems of the operators on CB[0,?)space.In chapter 4,we redefined ??-Bernstein type operators,analyzed the characteris-tics of these operators,then combine the differential mean value theorem,the Holder inequality,and then get the approximation direct,inverse and equivalence theorems of the operators in C[0,1]space.