Moduli of Smoothness for Weighted Lp ,0 < p < 1 Approximation

Abstract

Moduli of smoothness are intended for mathematicians working in approximation theory, numerical analysis and real analysis. Measuring the smoothness of a function by differentiability is too crude for many purposes in approximation theory. For many purposes many authors defined many types of moduli of smoothness. For easier work and for weighted approximation, we introduce some types of moduli of smoothness. In this paper we introduce a new moduli of smoothness for functions ƒ Lp [-1,1] n Cr_1 (-1,1), 0 < p < 1, r ≥ 1, that have an (r — 1)st locally bsolutely cotinous derivative in (-1,1), and such that φrfris in Lp [-1,1], where φ(x)= (1 - x 2)1/2. These moduli of smoothness are used to describe the smoothness of the derivative of functions in Lp spaces for 0 < p < 1, also it is equivalent to specific “weighted Ditzian-Totik (DT) moduli” of smoothness, but our definition is more straightforward and easier. we also introduce some relate properties with some relations and we prove the equivalence to “K- functional”