On equivalence of moduli of smoothness of splines in Lp,0<p<1Lp,0<p<1

It is shown that, if n,r∈Nn,r∈N, k∈N0k∈N0, 1⩽ν⩽r1⩽ν⩽r, tn≔cos(n-i)πni=0n is the Chebyshev partition of [-1,1][-1,1], and ss is a piecewise polynomial of degree ⩽r⩽r on tntn such that s∈Cν-1[-1,1]s∈Cν-1[-1,1], then for any 00t>0,ωk+νϕ(s,t)p⩽ctνωk,νϕ(s(ν),t)p,where ωk+νϕ and ωk,νϕ denote the Ditzian–Totik (k+ν)(k+ν)th modulus of smoothness and kkth modulus with the weight ϕνϕν, respectively. In particular, in the case k=0k=0, ωνϕ(s,t)p⩽c(r,p)tνϕνs(ν)p. It is known that these inequalities are no longer valid for a general ff in place of ss if 0