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

In metric of spaces $L_s,1<s\le \mathrm{\infty }$, 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 ${\mathrm{\Psi }}_{\beta }(t)={\sum }_{k=1}^{\mathrm{\infty }}\psi (k)\cos (kt-\frac{\beta \pi }{2})$, $\beta \in \mathbb{R}$, whose coefficients $\psi (k)$ are such that product $\psi (n)n^{1-\frac{1}{s}}$, $1<s\le \mathrm{\infty }$, can’t tend to nought faster than every power function and besides, if $1<s<\mathrm{\infty }$, then ${\sum }_{k=1}^{\mathrm{\infty }}{\psi }^s(k)k^{s-2}<\mathrm{\infty }$ and if $s=\mathrm{\infty }$, then ${\sum }_{k=1}^{\mathrm{\infty }}\psi (k)<\mathrm{\infty }$.