Order Estimates for the Best Approximations and Approximations by Fourier Sums in the Classes of Convolutions of Periodic Functions of Low Smoothness in the Uniform Metric

Abstract

We obtain the exact-order estimates for the best uniform approximations and uniform approximations by Fourier sums in the classes of convolutions of periodic functions from the unit balls of the spaces $L_p, 1 ≤ p < ∞$, with generating kernel $Ψ_{β}$ for which the absolute values of its Fourier coefficients $ψ(k)$ are such that $∑_{k = 1}^{∞} ψ_p ′(k)k^{p ′ − 2} < ∞,\; \frac 1p + \frac 1{p′} = 1$, and the product $ψ(n)n^{1/p}$ cannot tend to zero faster than power functions.