We consider the deviations of Fourier sums in the spaces \( {C^{\bar{\psi}}} \). The estimates of these deviations are expressed via the best approximations of the \( \bar{\psi} \) -derivatives of functions in the Stepanets sense. The sequences \( \bar{\psi} \) = (ψ1, ψ2) are quasiconvex.
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References
H. Lebesģue, “Sur la représentation trigonometrique approcheé des fonctions satisfaisant a une condition de Lipschitz,” Bull. Soc. Math. France, 38, 184–210 (1910).
V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
K. I. Oskolkov, “On the Lebesgue inequality in a uniform metric and on a set of complete measure,” Mat. Zametki, 18, No. 4, 515–526 (1975).
A. I. Stepanets, Approximation of \( \bar{\psi} \) -Integrals of Periodic Functions by Fourier Sums [in Russian], Preprint No. 96.11, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1996).
A. I. Stepanets, “Rate of convergence of Fourier series on the classes of \( \bar{\psi} \) -integrals,” Ukr. Mat. Zh., 49, No. 8, 1069–1113 (1997); English translation: Ukr. Math. J., 49, No. 8, 1201–1251 (1997).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
A. I. Stepanets, “Approximation of \( \bar{\psi} \) -integrals of periodic functions by Fourier sums (low smoothness). I,” Ukr. Mat. Zh., 50, No 2, 274–291 (1998); English translation: Ukr. Math. J., 50, No 2, 314–333 (1998).
A. I. Stepanets, “Approximation of \( \bar{\psi} \) -integrals of periodic functions by Fourier sums (low smoothness). II,” Ukr. Mat. Zh., 50, No. 3, 388–400 (1998); English translation: Ukr. Math. J., 50, No. 3, 442–454 (1998).
A. Kolmogoroff, “Sur l’ ordre de grandeur des coefficients de la serie de Fourier-Lebesģue,” Bull. Acad. Pol., Sér. Sci. Math., 83–86 (1923).
S. A. Telyakovskii, “Some estimates for trigonometric series with quasiconvex coefficients,” Mat. Sb., 63(105), No. 3, 426–444 (1964).
S. A. Telyakovskii, “Estimate for the norm of a function via its Fourier coefficients convenient in the problems of approximation theory” Tr. Mat. Inst. Akad. Nauk SSSR, 109, 65–97 (1971).
R. M. Trigub, “Approximation of continuous periodic functions with bounded derivative by polynomials,” in: Theory of Mappings and Approximation of Functions [in Russian], Naukova Dumka, Kiev (1989), pp. 185–195.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 6, pp. 844–849, June, 2013.
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Zaderei, N.M., Zaderei, P.V. On the Lebesgue Inequality on Classes of \( \bar{\psi} \) -Differentiable Functions. Ukr Math J 65, 938–944 (2013). https://doi.org/10.1007/s11253-013-0830-6
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DOI: https://doi.org/10.1007/s11253-013-0830-6