Monotone and comonotone polynomial approximation revisited
Let the \(L_ p[-1,1]\) best approximation by nth degree monotone polynomials to a monotone function f be denoted by \(E^*_ n(f)_ p\). The author proves \(E^*_ n(f)_ p\leq C\omega^ 2_{\phi}(f,1/n)_ p,\) where \(\phi (x)=\sqrt{1-x^ 2}\) and \(\omega^ 2_{\phi}(f,t)_ p=\sup_{0<h\leq t}\| \Delta^ 2_{h\phi}f\|_{L_ p[-1,1]}.\) The result was proved for polynomial approximation by \textit{V. Totik} and the reviewer in ``Moduli of smoothness'' (1987; Zbl 0666.41001). Using Theorem 3.1.2 of the book, one observes that the result of \textit{Xianming Yu} [Acta Math. Sin., New Ser. 3, 315-326 (1987; Zbl 0637.41004)] and the present result are equivalent. The proof in the present paper is, however, somewhat clearer.
scientific article; zbMATH DE number 4142708
scientific article