Free-knot splines approximation of \(s\)-monotone functions (Q1424936)

Splines with free knots provide a most useful example of nonlinear approximations. In this paper, the approximation orders of free-knot splines are studied when so-called \(s\)-monotone functions are approximated by free-knot splines and their monotonicity is preserved (``shape-preserving approximation''). The \(s\)-fold monotonicity is defined \textit{via} certain divided differences. Two-sided estimates of best approximations are given and relations with single layer perceptrons are studied in this article, too.