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 N 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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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 274–291, February, 1998.
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Stepanets, A.I. Approximation of \(\bar \psi - integrals\) of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I. Ukr Math J 50, 314–333 (1998). https://doi.org/10.1007/BF02513454
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DOI: https://doi.org/10.1007/BF02513454