Best monotone approximation in \(L_{\infty}[0,1]\)
If f(x) is a Lebesgue measurable function on [0,1] and \(p>1\), let \(f_ p(x)\) be the unique best \(L_ p\)-approximant to f(x) by non-decreasing functions on [0,1]. If \(\lim_{p\to \infty}f_ p(x)\equiv f_{\infty}(x)\) exists a.e., then \(f_{\infty}(x)\) is a best \(L_{\infty}\)-approximant to f(x) by non-decreasing functions. In this case we say that the Polya algorithm converges and \(f_{\infty}(x)\) is a best best \(L_{\infty}\)-approximant. Darst and Sahab have shown that if f(x) is quasi-continuous, then the Polya algorithm converges. Darst, Legg and Townsend have shown that if f(x) is only assumed Lebesgue measurable, then the algorithm may fail to converge. In this paper, we show that the condition that f(x) be quasi-continuous can be relaxed to the condition that f(x) can be uniformly approximated by simple measurable functions where one-sided limits need only exist at a few select points. The authors believe the construction of \(f_{\infty}(x)\) as given in this paper gives a clearer picture of what \(f_{\infty}(x)\) is, even when f(x) is continuous.
scientific article; zbMATH DE number 3867915