Weighted moduli of smoothness of kk-monotone functions and applications

Let ωφk(f,δ)w,Lq be the Ditzian–Totik modulus with weight ww, MkMk be the cone of kk-monotone functions on (−1,1)(−1,1), i.e.,    those functions whose kkth divided differences are nonnegative for all selections of k+1k+1 distinct points in (−1,1)(−1,1), and denote E(X,Pn)w,q:=supf∈XinfP∈Pn‖w(f−P)‖LqE(X,Pn)w,q:=supf∈XinfP∈Pn‖w(f−P)‖Lq, where PnPn is the set of algebraic polynomials of degree at most nn. Additionally, let wα,β(x):=(1+x)α(1−x)βwα,β(x):=(1+x)α(1−x)β be the classical Jacobi weight, and denote by Spα,β the class of all functions such that ‖wα,βf‖Lp=1‖wα,βf‖Lp=1.In this paper, we determine the exact behavior (in terms of δδ) of supf∈Spα,β∩Mkωφk(f,δ)wα,β,Lq for 1≤p,q≤∞1≤p,q≤∞ (the interesting case being q−1/pα,β>−1/p (if p<∞p<∞) or α,β≥0α,β≥0 (if p=∞p=∞). It is interesting to note that, in one case, the behavior is different for α=β=0α=β=0 and for (α,β)≠(0,0)(α,β)≠(0,0). Several applications are given. For example, we determine the exact (in some sense) behavior of E(Mk∩Spα,β,Pn)wα,β,Lq for α,β≥0α,β≥0.