The Polya algorithm in \(L_{\infty}\) approximation (Q789649)

Two aspects of the Polya algorithm are considered - (a) the monotone approximation in \(L_{\infty}(I)\) and (b) approximating functions in \(L_{\infty}(\Omega,{\mathcal A},\mu;x)\) by functions in \(L_{\infty}(\Omega,{\mathcal B},\mu;x)\) where \({\mathcal B}\) is sub \(\sigma\)- algebra of \({\mathcal A}\) and x is a uniformly convex Banach space. It is shown (a) how the Polya algorithm fails to converge for \(h\in L_{\infty}(I)\) and (b) how to find the best best \(L_{\infty}\) approximation to f.