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

Abstract

We obtain in the metric of spaces $L_{s}$, $1< s\leq\infty$, the exact order estimates of the best approximations and approximations by Fourier sums of classes of convolutions periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $\Psi_{\beta}(t)=\sum_{k=1}^{\infty}\psi(k)\cos\left(kt-\frac{\beta\pi}{2}\right)$, $\beta\in\mathbb{R}$. The kernel's coefficients $\psi(k)$ are such that product $\psi(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, tends to zero not faster than an arbitrary power function and if $1<s<\infty$, then $\sum_{k=1}^{\infty}\psi^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum_{k=1}^{\infty}\psi(k)<\infty$