The Study Of Some Approximation Problems In Ba Spaces

The approximation theory of function is one of the important Branches of modern mathematics. It began in the 19 th century when two famous theore -ms 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 approx -imation by polynomials was obtained by Chebyshev in 1859. It grew prosp -erous in 20th century and became an independent subject. People have gener -ated a series of theories and methods in order to use the simple and computa -ble functions to approximate general functions. For example, the best approxi -mation, Fourier approximation, approximation by trigonometric polynomials, approximation by algebraic polynomials, linear operators approximation, inter -polation approximation, rational approximation, approximation by reciproca -ls of polynomials, Müntz approximation and so on. With the appearance of complicated problems and nonlinear problems , studied this subject in wider spaces became inevitable. This article studied some problems in Ba spaces which was the generalization of Lp spaces. This article contains four chapters.Chapter one Preliminary knowledge In this section, we introduce the definition and properties of the Ba spaces.Chapter two Approximation by Beta operator in Ba spacesLinear operator approximation was fully developed in the 20th century, also gradually formed a whole set of mature theories. Beta operator is a kind of very important operators which were widely applied to approximation theory.In the first part of the second chapter ,we proved the equivalent of the K -functional and D-T modulus of smoothness, by means of the equivalent of them, we obtained the approximation characterizations of Beta operator for a kind of functions in Ba spaces which were the generalization of the theorems in document [4],[6].In the second part of the second chapter ,we discussed the approximation problems of Jacobi-weighted Beta operator in Ba spaces, also obtained the direct and inverse results﹑the characterizations of the approximation rate of Jacobi-weighted Beta operator and its derivative in Ba spaces on the bases of document [8],[9].Chapter three Interpolation approximation in Ba spacesIn the first part of chapter three, we consulted the methods of document [15],[17], further we modified a kind of even triangle interpolation polyno mial operators, and discussed the convergence and the degree of the convergence of the modified operator in continuous function spaces. Then, we esti mated the rate of convergence of the modified even triangle interpolation po -lynomials in Ba spaces.The results on Grünwald interpolation operator were numerous, most of them are based on the zero points of Chebyshev polynomials of the first or the second kind. In the second part of Chapter three, by means of the relationship between Lp(1n}^∞_n=1 was a monotonically increase sequence of real numbers,λ_n+1-λ_n≥M_n,the approximation problem of Sobolev function classes W¹_p[0,1]in Lp (p>1) spaces, and the approximation problem of bounded variation function classes BV_[0,1] in L1 spaces in document [32]. In this section, we discussed that whenΛ={λ_n}^∞_n=1 was a monotonically increase sequence of real numbers,λ_n+1-λ_n≥M_nα, 12≤α<+∞,the approximation problems of Sobolev function classes W p1[ 0,1]and bounded variation function classes BV_[0,1] in Ba_[0,1] spaces, also we got two approximation theorems which generalized the results in document [32].