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.
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References
Z. Ditzian and V. Totik,Moduli of Smoothness, Springer, New York, etc. (1987).
L. G. Shakh, “On the equivalence of some definitions ofkth moduli of continuity,” in:Fourier Series. Theory and Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 154–160.
V. K. Dzyadyk,Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
I. A. Shevchuk,Approximation by Polynomials and Traces of Continuous Functions Defined on Segments [in Russian], Naukova Dumka, Kiev (1992).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1627–1638, December, 1995.
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Dyuzhenkova, O.Y. A remark concerning the modulus of smoothness introduced by Ditzian and Totik. Ukr Math J 47, 1858–1872 (1995). https://doi.org/10.1007/BF01060960
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DOI: https://doi.org/10.1007/BF01060960