Abstract
In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order log N, where N−1 is the approximation step.
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S. V. Bochkarev, “On the absolute convergence of Fourier series,” Dokl. Akad. Nauk SSSR,202, No. 5, 225–227 (1972).
S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series [Russian translation], Moscow (1958).
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Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 363–370, March, 1974.
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Bochkarev, S.V. Absolute convergence of Fourier series with respect to bounded systems. Mathematical Notes of the Academy of Sciences of the USSR 15, 207–211 (1974). https://doi.org/10.1007/BF01438371
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DOI: https://doi.org/10.1007/BF01438371