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Uniform Approximations by Fourier Sums on the Sets of Convolutions of Periodic Functions of High Smoothness

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Ukrainian Mathematical Journal Aims and scope

On the sets of 2π-periodic functions f specified by the (ψ, β)-integrals of functions φ from L1, we establish Lebesgue-type inequalities in which the uniform norms of deviations of the Fourier sums are expressed via the best approximations of the functions φ by trigonometric polynomials in the mean. It is shown that obtained estimates are asymptotically unimprovable in the case where the sequences ψ (k) approach zero faster than any power function. In some important cases, we establish asymptotic equalities for the exact upper bounds of the uniform approximations by Fourier sums in the classes of (ψ, β)-integrals of the functions φ that belong to the unit ball in the space L1.

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  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    A. Serdyuk & T. Stepanyuk

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  1. A. Serdyuk
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Correspondence to T. Stepanyuk.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 4, pp. 542–567, April, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i4.7411.

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Serdyuk, A., Stepanyuk, T. Uniform Approximations by Fourier Sums on the Sets of Convolutions of Periodic Functions of High Smoothness. Ukr Math J 75, 621–651 (2023). https://doi.org/10.1007/s11253-023-02220-8

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  • DOI: https://doi.org/10.1007/s11253-023-02220-8

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