Modified Bernstein function and a uniform approximation of some rational fractions by polynomials

P. L. Chebyshev posed and solved (1857, 1859) the problem of finding an improper rational fraction least deviating from zero in the uniform metric on a closed interval among rational fractions whose denominator is a fixed polynomial of a given degree $m$ that is positive on the interval and numerator is a polynomial of a given degree $n\ge{m}$ with unit leading coefficient. A. A. Markov solved (1884) a similar problem in the case when the denominator is the square root of a given positive polynomial. In the 20th century, this research direction was developed by S. N. Bernstein, N. I. Akhiezer, and other mathematicians. For example, in 1964 G. Szegő extended Chebyshev's result to the case of trigonometric fractions using the methods of complex analysis. In this paper, using the methods of real analysis and developing Bernstein's approach, we find the best uniform approximation on a period by trigonometric polynomials of certain order for an infinite series of proper trigonometric fractions of a special form. It turned out that, in the periodic case, it is natural to formulate some results in terms of the generalized Poisson kernel $\Pi_{\rho,\xi}(t)=(\cos\xi)P_\rho(t)+(\sin\xi)Q_\rho(t)$, which is a linear combination of the Poisson kernel $P_\rho(t)=(1-\rho^2)/[2(1+\rho^2-2\rho\cos{t})]$ and the conjugate Poisson kernel $Q_\rho(t)=\rho\sin{t}/(1+\rho^2-2\rho\cos{t})$, where $\rho\in(-1,1)$ and $\xi\in\mathbb{R}$. We find the best uniform approximation on a period by the subspace $\mathcal{T}_{n}$ of trigonometric polynomials of order at most $n$ for the linear combination $\Pi_{\rho,\xi}(t)+(-1)^{n}\Pi_{\rho,\xi}(t+\pi)$ of the generalized Poisson kernel and its shift. For $\xi=0$, this yields Bernstein's known results on the best uniform approximation on $[-1,1]$ of the fractions $1/(x^2-a^2)$ and $x/(x^2-a^2)$ by algebraic polynomials. For $\xi={\pi}/{2}$, we obtain the weight analogs (with weight $\sqrt{1-x^2}$) of these results. In addition, we find the value of the best uniform approximation on a period by the subspace $\mathcal{T}_{n}$ of a special linear combination of the mentioned Poisson kernel $P_\rho$ and the Poisson kernel $K_\rho$ for the biharmonic equation in the unit disk.