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Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions

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Ukrainian Mathematical Journal Aims and scope

We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes C ψ β, p of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces L p , 1 ≤ p < ∞, with generating fixed kernels Ψβ ⊂ L p′, \( \frac{1}{p}+\frac{1}{{p^{\prime}}}=1 \), whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the L p -metric, 1 < p ≤ ∞, for the classes L ψ β,1 of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψβ ⊂ L p with functions from the unit ball in the space L 1. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.

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Authors and Affiliations

  1. Lesya Ukrainka West-European National University, Lutsk, Ukraine

    U. Z. Hrabova

  2. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    A. S. Serdyuk

Authors
  1. U. Z. Hrabova
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  2. A. S. Serdyuk
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 9, pp. 1186–1197, September, 2013.

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Hrabova, U.Z., Serdyuk, A.S. Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions. Ukr Math J 65, 1319–1331 (2014). https://doi.org/10.1007/s11253-014-0861-7

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  • DOI: https://doi.org/10.1007/s11253-014-0861-7

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