Sharp converse theorem of \(L^ p\)-polynomial approximation
This paper centers on a proof of a generalization of a converse Marchaud's inequality to the situation of \(L^ p\)-polynomial approximation on \([-1,1]\) using `weighted' moduli of continuity. The inequality, connecting different order (ordinary) moduli of smoothness was extended to \(L^ p[0,2\pi]\) approximation using trigonometric polynomials by Zygmund and Timam. Just as in their result, the main result in the paper under review connecting the weighted modulus of continuity \(\omega^ r_{\phi}(f,1/n)_ p\), \(\phi(x)=\sqrt{1-x^ 2},\) \(1<p<\infty\) to the p-th root of a sum of p-th powers of constants of best \(L^ p\)-approximation, shows quite different behaviour in the cases \(1<p\leq 2\) and \(2<p<\infty\). The second main theorem looks at other weights (satisfying certain boundedness conditions) and gives an extension of the Zygmund-Timam type integral inequality for \(1<p\leq 2\). Furthermore it is shown that this second theorem also holds for infinite intervals (the positive real axis is chosen), using the same method of proof. Finally it is pointed out that many other approximation processes can be treated to lead to Zygmund type inequalities by composition of the second main theorem and a Marchaud type result. As an example the author shows a strong converse result using Kantorovich polynomials on (0,1). The proofs are intricate and the second main theorem needs a cutting technique, applying a result by \textit{Z. Ditzian} [Proc. Am. Math. Soc. 103, 198-202 (1988)] on subintervals and pasting the results together in an ingenious way.
scientific article; zbMATH DE number 4127812