Study Of Several Operators Approximations And Polynomial Approximation Problems

Function approximation theory is one of the important branches of modern mathematics,which is an important part of function theory,and the basic problem involved is the problem of approximate representation of functions.The discipline derives from the feature theorem of best approximation proposed by Soviet mathematician Chebyshev in 1859 and the famous theorem established by German mathematician Weierstrass in 1885 that continuous functions can be approximated by polynomials.In the twentieth century,a series of in-depth work by Jackson,Bernstein and the Soviet school of analysis led to the flourishing of function approximation theory as an independent discipline.Nowadays,function approximation theory has become one of the most active branches of function theory,and the vigorous development of science and technology and the widespread use of fast electronic computers have given its development a powerful stimulus.Many branches of modern mathematics,including abstract disciplines such as topology,functional analysis,and algebra in basic mathematics,as well as computational mathematics,mathematical equations,probability and statistics,and some branches of applied mathematics,are closely related to approximation theory.So far,a considerable number of scholars at home and abroad have been engaged in this field,and there have been many research results in continuous function spaces and Lp spaces,but there are not many research results on function approximation in Orlicz spaces.In this paper,different tools are used to study the approximation of several operators,the approximation problem of polynomials,and the approximation problems of operators and the cases of weighted polynomials in Orlicz space.This article is divided into four chapters:The first chapter introduces the definition and properties of Orlicz space.The second chapter studies the weighted approximation problems of various linear operators in Orlicz spaces,which is divided into three subsections:Section 2.1 mainly discusses the approximation problem of the operator plus Jacobi weights in space,and uses and related analysis techniques to obtain the equivalence theorem and feature characterization of the weighted approximation of the operator in space,and Section 2.2 mainly discusses the approximation problem in Orlicz space under the weighting of a new Baskakov-type operator.The correlation analysis technique is used to obtain the positive theorem of the weighted approximation of the operator in the Orlicz space,and Section 2.3 studies the convergence and approximation properties of the exponential weighted approximation of the modified Picard operator in the Orlicz space.By establishing the correlation lemma of exponential weighted approximation in Orlicz space,the positive theorem and related properties of the operator in Orlicz space are obtained by using Korovkin’s theorem,Jensen’s inequality of convex functions,Minkowski’s inequality and related analysis techniques.Chapter 3 studies the problem of polynomial-weighted approximation,divided into two subsections:Section 3.1 examines the best polynomial approximation problem in Orlicz spaces with Jacobi weights,and uses De la Vallee Poussin averages,Jensen inequalities for convex functions,inequalities,and related analytical techniques to give Jackson-Favard type estimates of polynomials with Jacobi weights in Orlicz spaces of no more than degree.Section 3.2 studies the problem of polynomial approximation with exponential weights w(x)=e-(1-x2)-α(α>0)in Orlicz spaces,and proves Jackson’s theorem and its weak form in Orlicz spaces by introducing new smooth modes and related K-functionals,uses inequalities and correlation analysis techniques,and obtains a new Bernstein inequality,on the basis of which the best approximation theorem for polynomial derivatives is given.