On moduli of smoothness ofk-monotone functions and applications

Abstract

AbstractLetkbe the set of allk-monotone functions on (−1, 1), i.e., those functionsffor which thekth divided differences [x0,. . .,xk;f] are nonnegative for all choices of (k+1) distinct pointsx0,. . .,xkin (−1, 1). We obtain estimates (which are exact in a certain sense) ofkth Ditzian–Totikq-moduli of smoothness of functions ink∩p(−1, 1), where 1 ≤q<p≤ ∞, and discuss several applications of these estimates.