Research On Several Approximation Problem Of Operators

As an important part of modern mathematics,function approximation theory was originally founded by Weierstrass and Chebyshev.In 1859,Chebyshev proposed the characteristic theorem of the best approximation,and in 1885,Weierstrass established the famous theorem of polynomial approximation.Since then,under the special research of Jaskson,Bernstein and Soviet scholars,function approximation theory has become an independent discipline and gradually become an important part of modern mathematics,mainly including operator approximation,interpolation approximation,Fourier approximation,width and other related problems.Up to now,many scholars at home and abroad have studied problems in continuous function space and L_Pspace,and obtained a lot of classical results.With the emergence of more and more complex problems in the research process,it is urgent to study problems in a relatively wider space.Because the topological structure of Orlicz space is more complex than that of L_Pspace,the discussion of problems in Orlicz space has certain expanding significance and application prospects.This paper mainly studies a series of operator approximation in Orlicz space,mainly including univariate operator and multivariate operator approximation.The full text is divided into three chapters.The first chapter Some basic contents in Orlicz space.The second chapter Univariate operator approximation.The approximation properties of univariate operators in Orlicz space are studied.This chapter is divided into four sections.In the first section,the weighted approximation properties of Gauss-Weierstrass operator in Orlicz space are explored.By using the equivalence between K-functional and smooth moduli and Jensen inequality of convex function,the equivalence theorem of approximation of the operator in Orlicz space is obtained.In the second section,the related approximation properties of the modified Baskakov type operator are investigated in Orlicz space,and the strong inverse inequality of this kind of operator is constructed in Orlicz space by using the tools of Holder inequality,K-functional and Cauchy inequality.In the third section,the approximation properties of Bernstein-Kant-orovich quasi-interpolants are explored in Orlicz space.The boundedness of the operator is obtained by calculating the complex norm in Orlicz space.By using the tools of Hard-Littlewood function,Jensen inequality and K-functional,the positive theorem,inverse theorem and equivalent theorem of the operator are obtained in Orlicz space.In the fourth section,a class of Kantorovich type operator based on Fejer kernel is explored.The approximation degree of the operator in Orlicz space is estimated by using the inequality technique of convex function and K-functional.The third chapter Multivariate operator approximation.The approximation properties of multivariate operators in Orlicz space are studied.This chapter is divided into two sections.In the first section,the convergence of a binary Kantorovich type operator in Orlicz space is discussed.According to the definition of norm in Orlicz space,the boundedness of the binary Kantorovich type operator is calculated,and the positive theorem of approximation of the operator is obtained by using the properties of convex function and inequality techniques.In the second section,the approximation theorems of binary Bernstein Durrmeyer operator in Orlicz space are mainly introduced.According to the convexity of N function,Hardy Littlewood maximum function,Holder inequality and so on,a series of approximation properties of this kind of multivariate operator are obtained in Orlicz space.