One-sided integral approximations of the generalized Poisson kernel by trigonometric polynomials

We consider the generalized Poisson kernel $\Pi_{q,\alpha}=\cos(\alpha \pi/2)P +\sin(\alpha\pi/2)Q$ with $q\in(-1,1)$ and $\alpha\in\mathbb{R}$, which is a linear combination of the Poisson kernel $P(t)=1/2+\sum_{k=1}^\infty q^k\cos{kt}$ and the conjugate Poisson kernel $Q(t)=\sum\nolimits_{k=1}^\infty q^k\sin kt$. The values of the best upper and lower integral approximations of the kernel $\Pi_{q,\alpha}$ by trigonometric polynomials of order not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.