Jackson and smoothness theorems for freud weights

Abstract

We prove Jackson, realization, and converse theorems for Freud weights inL p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL p for 0<p ≤ ∞. Since theK-functional is identically zero inL p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.