The Study Of Some Operators Approximation And Interpolation Approximation Problems

The approximation theory of function is one of the important Branches of modern mathematics.It began in the 19th century when two famous theorems were established,one theorem said that the continuous functions can be approximated by certain polynomials which was established by Weierstrass in 1885,and the other which established characteristic theorems for best approximation by polynomials was obtain by Chebyshev in 1859.It grew prosperous in 20th century and became an independent subject.People have generated a series of theories and methods in order to use the simple and computable functions to approximate general functions. For example, the best approximation,Fourier approximation, approximation by trigonometric polynomials,approximation by algebraic polynomials, linear operators approximation,interpolation approximation, rational approximation approximation by reciprocals of polynomials, Muntz approximation and so on.With the appearance of complicated problems and nonlinear problems,studied approximation theories in wider spaces which is very important.This article studied operators approximation, interpolation approximation in Orlicz spaces and LMBa spaces which were composed of a series of Orlicz spaces. This article contains three chapters.Chapter one Preliminary knowledgeIn this section, we introduce the definition and properties of the Orlicz spaces and LMBa spaces.Chapter two Operators approximationThe chapter which mainly studies the approximation of two diff-erence types linear operators,and obtain corresponding approximation theorems.section one with the K-functional and modulus of smoothness as the tool,we discuss Durrmeyer-Bezier operators in LMBa spaces,and two estimates of degree of approximation are obtained. Section two with the K-functional as a tool establish Bernstein Inequality in Orlicz spaces,we discuss approximation theorem of Shepard operators in Orlicz spaces.Chapter three Interpolation approximationThe chapter which mainly introduce the question of interpolation approximationand obtain corresponding approximation theorems.Sec-tion one,With [2] the relevant conclusions,we studied the second kind Chebyshev polynomial interpolation nodes for the zeros of the quasi Grunwald interpolation polynomials in the weighted degree of approximation in Orlicz spaces.Section two,we studied Muntz Rational Approximation in Orlicz spaces and the Jackson type estimates of degree of approximation are obtained.Section three,we dicuss the first class of modified rational interpolation approximation in LM,ωBa spaces by means of the relation of Orlicz norm and LMBa norm,obtained degree of approximation in LM,ωBa spaces.