Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

Abstract

In this paper we pursue the study of the best approximation operator extended from L^Φ to L^φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calderón-Zygmund class t^p m(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska-Orlicz indices for the function φ.