Study On Several Approximation Problems In Orilcz Space

Function approximation theory is an important branch of modern mathematics,The problems studied mainly include operator approximation,rational approximation,interpolation approximation,polynomial approximation,best approximation,width theory and other related problems.There are many researches on these problems in continuous function space and Lp space,but relatively few in Orlicz space larger than Lp space.With the emergence of more and more complexity problems and nonlinear problems,It is a historical necessity to study the function approximation problem in Orlicz spacesIn this paper,we study operator approximation,interpolation approximation and Muntz-rational approximation in Orlicz spaces.The full text is divided into four chaptersThe first chapter preparatory knowledgeThis paper introduces the basic knowledge of Orlicz spaceThe second chapter operator approximationThe first Section By using Jensen's inequality,Hardy-Littlewoods maximal function,K-functional and integral continuous module equivalence,the convergence and approximation characteristics of a kind of recursive Kantorovich type operators in Orlicz spaces is studied,and the degree of approximation of the operator is obtainedThe second section Szasz-Durmeyer-Bezier operators is introduced and their approximation problem in Orlicz space is studied.the theorem of approximation of the operator is given by using Jensen inequality and common measure in function approximation theory in Orlicz spacesThe third chapter interpolation OperatorsIn the first section we mainly study the approximation problems of the Hermite interpolation operators which is based on the zeros of Chebyshew polynomials of the second kinds in the Orlicz spaces.The degree of approximation of the interpolation operators is given by using the Holder inequality,Hardy-littlewoods maximal function,continuous modulus and the convexity of the N-functionin the Orlicz spaceIn the second section we construct a modified Kantorovich type weighted Grunwald interpolation operator and study the approximation problem of the interpolation operator in Orlicz spaces.We use the usual tools in functional approximation theory and related analysis techniques to obtain the approximation order of the interpolation operator in Orlicz spacesThe fourth chapter Muntz rational approximationCompare with previous studies on similar problems and tries to improve the index {?n}n=1 ?,using Holder inequality,Hardy-Littlewoods maximal function,K-functional continuous modulus and convexity of N-function techniques,the necessary conditions lead to a conclusion of Jackson theorem of approximation.