On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels

In the metrics $C$ and $L$ we solve the problem of best approximation by trigonometric polynomials in classes of continuous periodic functions $f(x)$ of the form
$$f(x)=\frac1n\int^{2\pi}_0K(t)\varphi(x-t)\,dt,$$
where the kernel $K(t)$ is a periodic integral of a linear combination of functions that are absolutely monotonic in the intervals $(-\infty,2\pi)$ and $(0,\infty), and $\|\varphi\|\le1$.
A~particular case of such kernels for any $s>0$ and $\alpha\in(-\infty,+\infty)$ are kernels of the form $$K(t)=\sum^\infty_{k=1}\frac{\cos(kt-\frac{\alpha\pi}2)}{k^s},$$ which for $\alpha=s$ generate classes of periodic functions with a bounded $s$-th derivative in the sense of Weyl, whereas for $\alpha=s+1$ they generate conjugate classes. For various values of $s$ and $\alpha$, apart from the case $s\in(0,1)$ and $\alpha\in[0,2]\setminus[s,2-s]$, such kernels were studied by various investigators (see [1-?12]).