Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics

Abstract

In metric of spaces $L_s,1<s\le \infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_1$, with generating kernel ${\Psi }_{\beta }(t)=\sum _{k=1}^{\infty }\psi (k)\cos (kt-\frac{\beta \pi }{2})$, $\beta \in \mathbb{R}$, whose coefficients $\psi (k)$ are such that product $\psi (n)n^{1-\frac{1}{s}}$, $1<s\le \infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum _{k=1}^{\infty }{\psi }^s(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum _{k=1}^{\infty }\psi (k)<\infty$.

23 pages, in Ukrainian