Three-monotone spline approximation
Jackson theorems for functions \(f\) whose third order divided differences are always nonnegative are studied. They are approximated by piecewise polynomial splines with minimal defect, and the error is bounded above in a Jackson-type theorem using the fourth modulus of smoothness of \(f\). Such functions are called three-monotone, and so is the spline which is used as an approximation. The error-estimate is in the uniform norm. Analogous results are provided using the Ditzian-Totik modulus of smoothness. They also hold in the uniform norm (and not in other \(p\)-norms, \(p\) finite, so long as the knots of the approximating spline are independent of \(f\)).
3-monotone approximation by piecewise polynomials and splines
scientific article; zbMATH DE number 5840441
scientific article