Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions
Abstract
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.
English version (Springer): Ukrainian Mathematical Journal 65 (2013), no. 9, pp 1319-1331.
Citation Example: 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. Mat. Zh. - 2013. - 65, № 9. - pp. 1186–1197.
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