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

  • U. Z. Hrabova
  • A. S. Serdyuk

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.
Published
25.09.2013
How to Cite
HrabovaU. Z., and SerdyukA. S. “Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, no. 9, Sept. 2013, pp. 1186–1197, http://umj.imath.kiev.ua/index.php/umj/article/view/2500.
Section
Research articles