Research On Some Approximation Problems In Orlicz Space

As an important branch of modern mathematics,function approximation theory first emerged in the former Soviet Union.In 1859,the famous Soviet mathematician Chebyshev proposed the characteristic theorem of the best approximation.In 1885,the German mathematician Weierstrass proved that continuous functions can always be approximated by polynomials.Later,with the dedicated research of a large number of mathematicians such as Jackson and Bernstein,function approximation theory,as an independent branch of the field of analysis,has been developed in mathematical theoretical research and practical application.With the development of science and technology,the connection between function approximation theory and other disciplines is also deepening.At present,the research system of operator approximation,rational approximation,interpolation approximation and width problems in continuous function space and Lp space has been relatively complete,and Orlicz space,as a function space larger than Lp space,has more research significance.Extensive,especially the Orlicz space generated by the N function that does not satisfy the ?2 condition is a substantial extension of the Lp space,so the study of the approximation problem in the Orlicz space has more extended significance.The full text is divided into five chapters:The first chapter introduces the definition and properties of Orlicz space and width,and gives the corresponding notation.The second chapter studies the problem of operator approximation in Orlicz space.This chapter is divided into three sections:the first section studies the approximation equivalence of the modified Polyá operator in Orlicz space with the help of Hardy-Littlewood maximal functions,Jensen's inequality of convex functions,and K-functional and continuous modes in Orlicz space;in the second section,the saturation problem of a class of mixed exponential integral operators in Orlicz space is discussed,and this kind of mixed exponential integral is given by constructing a bilinear functional.The saturation theorem of operators in Orlicz space;the third section introduces the multivariate Baskakov-Durrmeyer operator,and uses tools such as K-functional,smooth modulus,and dimension decomposition to establish a strong direct approximation of operators in Orlicz space inequality.The third chapter mainly discusses the problem of rational approximation.For functions that change the sign l times in[-1,1]in Orlicz space,it is proved that f can be approximated by rational functions in Rnl,and the corresponding Jackson type theorem is established.The fourth chapter explores the problem of interpolation approximation.For the triangular polynomial with equidistantly distributed interpolation nodes,this chapter establishes the asymptotic equation for the approximation of the interpolating triangular polynomial in Orlicz space by means of the generalized Minkowski inequality,and gives the theorems for different function classes under the three conditions in the Orlicz space.The fifth chapter studies the problem of width in Orlicz space.This chapter is divided into two sections:the first section studies the asymptotically accurate estimation of the Kolmogorov width in L1 and the asymptotically optimal subspace of the aperiodic function class in Orlicz whose spatial domain is[-?,?],and further discusses the Kolmogorov Width,linear width,and dual of Gelfand width;the second section considers the relationship between Kolmogorov width,linear width and Gelfand width in Orlicz space for a class of periodic smooth functions in Lp space and periodic functions constrained by certain types of boundary conditions.