A Note on Convex Approximation in Lp

✍ Scribed by M. Nikoltjevahedberg; V. Operstein

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 116 KB
• Volume: 81
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1039

✦ Synopsis

A convex function (f) given on ([-1,1]) can be approximated in (L_{r}, 1<p<x). by convex polynomials (P_{n}) of degree at most (n) with the accuracy (o\left(n^{-2 i p}\right)). This follows from the estimate (\left|f-P_{n}\right|{p} \leqslant c \cdot n^{-2 / p} \cdot \omega{2}^{\varphi}\left(f, n^{-1}\right)^{1 / 4}). where (1 \leqslant p \leqslant \infty), (p^{-1}+q^{-1}=1, \varphi(x)=\left(1-x^{2}\right)^{1 / 2}), and (\omega_{2}^{\varphi}(f, t)) is the Ditzian-Totik modulus of smoothness in the uniform metric. Of 1995 Academic Press. Inc