We prove Jackson, realization, and converse theorems for Freud weights inLp, 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 inLp for 0<p ≤ ∞. Since theK-functional is identically zero inLp,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inLp for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.
Primary 41A10 42C05
Key words and phrases
Freud weights Polynomial approximation Jackson theorems Moduli of continuity K-Functionals Realization
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