Research On Approximation Problem Of Several Functions

Study on the problems of approximation theory of functions including linear operator approximation,interpolation approximation,rational approximation,approximation by trigonometric polynomials,polynomial approximation,the width of the relevant problems.These problems have many research in continuous function space and space,the research in Orlicz space is relatively small,and the Orlicz space the space is covered,so in the Orlicz space of the function approximation problem has a certain academic value.This paper is divided into five chapters:the knowledge of preparation,approximation of Lupas-Baskakov operator,approximation of several Kantorovich interpolation operator,approximation of Müntz rationalfunctions and the optimal subspace of then-Kwidth of an important function class.The first chapter introduces the knowledge and some marks of the Orlicz space.The second chapter studies the approximation of Lupas-Baskakov operators in Orlicz space,this chapter is divided into two parts,the first part of the study of Lupas-Baskakov type operators in Orlicz space boundedness,convergence,using continuous mode,Hardy-littlewood maximal function and Jensen inequality are given to estimate the degree of approximation operator in Orlicz space weighted sense that is,the inverse theorem of approximation operator in Orlicz spaces.The second part discusses the approximation properties of Lupas-Baskakov operators in Orlicz space,the use of Ditzian-Totik mode and H?lder inequality tool to obtain the strong type of this operator in Orlicz spaces approximation inverse theorem.The third chapter studies the approximation of some Kantorovich type interpolation operator in Orlicz space.This chapter is divided into two parts,the first part of the Hermite interpolation operator of Kantorovich type amended,weighted approximation properties of Hermite interpolation operator in Orlicz space.In the second part,five kinds of Bernstein interpolation operators are introduced.After these Kind of Bernstein interpolation operators are modified by appropriate Kantorovich type,we study the weighted approximation properties of the modified Bernstein interpolation operators in Orlicz space.The boundedness of these five Kantorovich operators in Orlicz space is given.The approximation estimates of five kinds of operators in Orlicz space are given by using the smoothing modulus,Hardy-Littlewood maximal function and convexity of the N function.The fourth chapter studies the Müntz rational approximation in Orlicz space.The chapter is divided into two parts.The first part studies the approximation property of Müntz rational function in the weighted Orlicz space.The Hardy-Littlewood maximal function and the convexity of the function are given In the Orlicz space,the pointwise estimator and the global estimator of the rational function are introduced.The second part introduces a modified Kantorovich-Bak operator,studies the weighted approximation degree estimation of the operator in Orlicz space,and gives the approximation degree estimate of the rational function in Orlicz space by means of weighted continuous modules and weighted K functionals.In the fifth chapter,we study the polar subspace problem of r order generalized function classW_M^rinL₁space.By using the equivalence between Orlicz norm and Luxemburg norm in Orlicz space and the definition of Kolmogorov width,we obtain the polar subspace of the generalized function class in space.