A remark concerning the modulus of smoothness introduced by Ditzian and Totik

Abstract

For each functionf(x) continuous on the segment [−1, 1], we set \(\tilde f(t) = f(\cos t)\) . We study the relationship between the ordinarykth modulus of continuity \(\omega _k (\tau ,\tilde f^{(r)} )\) of therth derivative \(\tilde f^{(r)}\) of the function \(\tilde f\) and thekth modulus of continuity \(\bar \omega _{k,r} (\tau ,f^{(r)} )\) with weight ϕ _r of the rth derivativef ^(r) of the functionf introduced by Ditzian and Totik. Thus, ifr is odd andk is even, we prove that these moduli are equivalent ast→0.