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

  • O. Yu. Dyuzhenkova

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.
Published
25.12.1995
How to Cite
DyuzhenkovaO. Y. “A Remark Concerning the Modulus of Smoothness Introduced by Ditzian and Totik”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 47, no. 12, Dec. 1995, pp. 1627–1638, https://umj.imath.kiev.ua/index.php/umj/article/view/5556.
Section
Research articles