^{Of Variable Index L P (·) Space Some Approximation Problems}

In this paper, we mainly consider approximation problems in variable Lp(·) spaces. We first establish approximation theorems on variable Lp(·) spaces of the d-dimensional space with multivariate positive linear operators. Next we intro-duce definitions of the K-functional and the moduli of smoothness on variable Lp(·) spaces, respectively. And then we give the equivalence relation between them. This paper is arranged as follows.In chapter1, we present the background of variable LP(·) spaces, introduce the K-functional and the moduli of smoothness on variable LP(·) spaces and de-scribe some important theorems in this paper.In chapter2, for arbitrary (ε,∞)-domain Ω in Rd, we prove the approxima-tion theorem of multivariate positive linear operators in the LP(·)(Ω) space when the exponent function p:Ωâ†'[1,∞) satisfies d<p-≤p+<∞and the log-Holder continuity. As applications, we give approximation estimates of the gen-eralized multivariate Bernstein-Durrmeyer operators and Bernstein-Kantorovich operators on simplex S∈Rd, respectively.In chapter3, we prove some properties of the moduli of smoothness on vari-able LP(·)(Ω)(Ω∈Rd) spaces by using the boundedness of the Hardy-Littlewood maximal operator. Then we discuss the relationship between the K-functional and the moduli of smoothness.In chapter4, we study properties of the moduli of smoothness on variable Lp(·)([0,1]) spaces via the uniform boundedness of the Steklov convolution oper-ator family. And then we characterize the equivalence between the K-functional and the moduli of smoothness according to these properties.