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 Ψ_β ⊂ 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 Ψ_β ⊂ 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.