On rational approximations of functions and eigenvalue selection in Werner algorithm

The paper deals with the best uniform rational approximations (BURA) of continuous functions on compact (and even finite) subsets of real axis $\mathbb R$. The authors show that BURA does not always exist. They study the algorithm of Helmut Werner in more detail. This algorithm serves to search for BURA of the type $P_m/Q_n=\sum_{i=0}^ma_ix^i\big/\sum_{j=0}^nb_jx^j$ for functions on a set of $N=m+n+2$ points $x_1<\dots<x_N$. It can be used within the Remez algorithm of searching for BURA on a segment. The Werner algorithm calculates $(n+1)$ real eigenvalues $h_1,\dots,h_{n+1}$ for the matrix pencil $A-hB$, where $A$ and $B$ are some symmetric matrices. Each eigenvalue generates a rational fraction of the type $P_m/Q_n$ which is a candidate for the best approximation. It is known that at most one of these fractions is free from poles on the segment $[x_1,x_N]$, so the following problem arises: how to determine the eigenvalue which generates the rational fraction without poles? It is shown that if $m=0$ and all values $f(x_1),-f(x_2),\dots,(-1)^{n+2}f(x_{n+2})$ are different and the approximating function is positive (negative) at all points $x_1,\dots,x_{n+2}$, then this eigenvalue ranks $[(n+2)/2]$-th ($[(n+3)/2]$-th) in value. Three numerical examples illustrate this statement.