Abstract
We investigate the rate of convergence of Fourier series on the classes \(L^{\bar \psi } \) N in the uniform and integral metrics. The results obtained are extended to the case where the classes \(L^{\bar \psi } \) N are the classes of convolutions of functions from ℜ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets \(L^{\bar \psi } \) N, which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.
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A. I. Stepanets, “Approximation of \(\bar \psi - integrals\) of periodic functions by Fourier sums (small smoothness). I,” Ukr. Mat. Zh., 50, No. 2, 274–291 (1998).
A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials [in Russian], Naukova Dumka, Kiev (1981).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
S. M. Nikol’skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, No. 3, 207–256 (1946).
A. S. Demchenko, “Approximation of functions from the classes W r H [ω]L in the mean,” Ukr. Mat. Zh., 25, No. 2, 267–276 (1973).
A. I. Stepanets, “On the Lebesgue inequality on the classes of (ψ, β)-differentiable functions,” Ukr. Mat. Zh., 41, No. 5, 449–510 (1989).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 388–400, March, 1998.
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Stepanets, A.I. Approximation of \(\bar \psi - Integrals\) of periodic functions by Fourier sums (small smoothness). IIof periodic functions by Fourier sums (small smoothness). II. Ukr Math J 50, 442–454 (1998). https://doi.org/10.1007/BF02528808
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DOI: https://doi.org/10.1007/BF02528808