Research On The Approximation Properties Of Parametric Operators In Infinite Intervals

Approximation theory of functions is an important research direction in modern mathematics.It is closely related to other branches of mathematics.In 1951,Korovkin and Bohman combined the approximation theory with functional analysis,thus the study of operator approximation theory began.A large number of mathematicians are constantly studying in this field and have obtained rich results.In order to improve the approximation effect on the infinite interval[0,1),on the basis of previous studies,we construct Gamma type and Szász-Mirakyan type operators preserving functions e^2μxand e^μx（μ>0）,and further extend the approximation theorems of these two types of operators.This paper is divided into four parts.In Chapter 1,we give the concepts involved in this paper such as the space,norm,modulus,K-functional,as well as the inequalities and expansions.In Chapter 2,the definitions of the four kinds of parametric operators are given respectively.Next,we calculate the first,second and fourth order moments of these operators.We give some lemmas required for subsequent proof.We obtain the shape preserving properties of the Szász-type basis functions.In Chapter 3,we research the uniform approximation properties of the four types of operators with parameters.The direct theorem of uniform approximation and weak inverse theorem of Voronovskaja type of these four types of operators are proved.The approximation error is further characterized by different types of modulus.At the same time,by comparing the four types of operators constructed with the classic ones,we get the function types that are more effective to the newly constructed operators.In order to compare the approximation effect of the operators on different kinds of functions,we take Szász-Mirakyan type operator as an example,and choose some functions,and carry out graphs by means of Matlab mapping software,and calculate the root mean squared error of the approximation.In Chapter 4,we give the definitions of statistical convergence,study the properties of statistical approximation of the four types of newly constructed operators.The Korovkin type theorem in the sense of weighted matrix statistical convergence and the same order weighted matrix statistical convergence theorem are also obtained.