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

For each functionf(x) continuous on the segment –1, 1], we set . We study the relationship between the ordinarykth modulus of continuity of therth derivative of the function and thekth modulus of continuity with weight _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 ast0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1627–1638, December, 1995.