Sufficient condition for the best uniform approximation by simple partial fractions (Q354822)

Let \(f\) be a real-valued continuous function on \([-1,1]\). Under the assumption that a certain algebraic identity holds, the author shows that if \(R_n\) is a simple partial fraction of degree at most \(n\) whose poles are simple and are outside the unit disk then it is a best uniform approximation to \(f\) from the set of all simple partial fractions of degree at most \(n\), provided that \(f-R_n\) has a Chebyshev alternation of \(n+1\) points on \([-1,1]\). The result is applied to the problem of approximation of real constants. The algebraic identity mentioned above depends on \(n\). The present article contains a proof for \(n=3\), and the author points out that so far its validity has only been verified for \(n \leq 5\).