Polynomial approximation in \(L_ p(S)\) for \(p>0\)
For a simple polytope \(S\) in \(R^d\) and \(p>0\), the author shows that the best possible approximation \(E_n (f)_p\leq Cw^r_S (f,1/n)_p\) where \(w^r_S (f,1/n)_p\) is a measure of smoothness of \(f\). The result is the best possible in the sense that a weak-type converse inequality is shown and a realization of \(w^r_S (f,t)_p\) via polynomial approximation is proved. A weaker inequality with \(1\leq p\leq\infty\), was proved in \textit{Z. Ditzian} and \textit{V. Totik} (Moduli of smoothness (1987; Zbl 0666.41001), Chapter 12). The above inequality for \(1\leq p \leq\infty\) is proved elsewhere by Z. Ditzian and K. Ivanov (manuscript).
scientific article; zbMATH DE number 922393