We establish uniform (with respect to the parameter p, 1 ≤ p ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes C ψ β,p of periodic functions generated by sequences ψ(k) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function ψ. Similar estimations are obtained for the best approximations of the classes L ψ β,1 in metrics of the spaces L s , 1 ≤ s ≤ ∞.
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References
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
A. I. Stepanets, A. S. Serdyuk, and A. L. Shidlich, “Classification of infinitely differentiable functions,” Ukr. Mat. Zh., 60, No. 12, 1686–1708 (2008); English translation: Ukr. Math. J., No. 12, 1982–2005 (2008).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
A. S. Serdyuk, “One linear method for the approximation of periodic functions,” in: Problems of Approximation Theory of Functions and Related Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 1, No. 1 (2004), pp. 294–336.
A. S. Serdyuk, “On the best approximation on the classes of convolutions of periodic functions,” in: Approximation Theory and Its Applications, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 41 (2002), pp. 168–189.
A. S. Serdyuk and T. A. Serdyuk, “Order estimates for the best approximations and approximations of the classes of infinitely differential functions by Fourier sums,” Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 10, No. 1 (2013), pp. 255–282.
V. S. Romanyuk, “Supplements to the estimation of approximation of infinitely differentiable functions by Fourier sums,” in: Extremal Problems of the Theory of Functions and Related Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 46 (2003), pp. 131–135.
A. S. Serdyuk and I. V. Sokolenko, “Uniform approximations of the classes of (\( \psi, \overline{\beta} \))-differentiable functions by linear methods,” in: Approximation Theory of Functions and Related Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 8, No. 1 (2011), pp. 181–189.
A. S. Serdyuk and I. V. Sokolenko, “Approximations of the classes of (\( \psi, \overline{\beta} \))-differentiable functions by linear methods,” Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 10, No. 1 (2013), pp. 245–254.
A. S. Serdyuk and Ie. Yu. Ovsyi, “Uniform Approximation of Periodical Functions by Trigonometric Sums of a Special Type,” ISRN Math. Anal., 165389 (2014).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1962).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 9, pp. 1244–1256, September, 2014.
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Serdyuk, A.S., Stepanyuk, T.A. Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics. Ukr Math J 66, 1393–1407 (2015). https://doi.org/10.1007/s11253-015-1018-z
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DOI: https://doi.org/10.1007/s11253-015-1018-z