A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II

In the problem of approximating real functions $f$ by simple partial fractions of order ${\le}\,n$ on closed intervals $K=[c-\varrho,c+\varrho]\subset\mathbb{R}$, we obtain a criterion for the best uniform approximation which is similar to Chebyshev's alternance theorem and considerably generalizes previous results: under the same condition $z_j^*\notin B(c,\varrho)= \{z\colon|z-c|\le\varrho\}$ on the poles $z_j^*$ of the fraction $\rho^*(n,f,K;x)$ of best approximation, we omit the restriction $k=n$ on the order $k$ of this fraction. In the case of approximation of odd functions on $[-\varrho,\varrho]$, we obtain a similar criterion under much weaker restrictions on the position of the poles $z_j^*$: the disc $B(0,\varrho)$ is replaced by the domain bounded by a lemniscate contained in this disc. We give some applications of this result. The main theorems are extended to the case of weighted approximation. We give a lower bound for the distance from $\mathbb{R}^+$ to the set of poles of all simple partial fractions of order ${\le}\,n$ which are normalized with weight $2\sqrt x$ on $\mathbb{R}^+$ (a weighted analogue of Gorin's problem on the semi-axis).