A remark concerning the modulus of smoothness introduced by Ditzian and Totik - Dyuzhenkova, O. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 1995)
For each functionf(x) continuous on the segment [−1, 1], we set \(f(t) = f(t)\) . We study the relationship between the ordinarykth modulus of continuity \(ω _k (τ ,f^(r) )\) of therth derivative \(f^(r)\) of the function \(f\) and thekth modulus of continuity \(ω _k,r (τ ,f^(r) )\) 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.