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

AbstractIt is shown that, if n,r∈N, k∈N0, 1⩽ν⩽r, tn≔cos(n-i)πni=0n is the Chebyshev partition of [-1,1], and s is a piecewise polynomial of degree ⩽r on tn such that s∈Cν-1[-1,1], then for any 0<p<1 and t>0,ωk+νϕ(s,t)p⩽ctνωk,νϕ(s(ν),t)p,where ωk+νϕ and ωk,νϕ denote the Ditzian–Totik (k+ν)th modulus of smoothness and kth modulus with the weight ϕν, respectively. In particular, in the case k=0, ωνϕ(s,t)p⩽c(r,p)tνϕνs(ν)p. It is known that these inequalities are no longer valid for a general f in place of s if 0<p<1 even if it is assumed that f∈C∞[-1,1].This implies, in particular, that if a piecewise polynomial s of degree ⩽r on tn is such that s∈Cm[-1,1], 0⩽m⩽r-1, then for any 1⩽k⩽r+1, 1⩽ν⩽min{k,m+1} and 0<p