The algebraic fractions of Chebyshev and Markov on several segments
An algebraic fraction of Chebyshev-Markov on the interval \(F=[0,1]\) is of the form \[ {x^n+c_1 x^{n-1}+\ldots +c_n\over \prod_{i=1}^{2n}(1-a_{i,n}x)^{1/2}}, \] where the reals \(c_{1},\ldots,c_n\) are chosen such that the fraction least deviates from zero, i.e., \[ \Biggl\| {x^n+c_1x^{n-1}+\ldots +c_n\over \prod_{i=1}^{2n}(1-a_{i,n}x)^{1/2}}\Biggr\| _{C(F)} \rightarrow \min_{\{c_1,\ldots ,c_n\} \subset {\mathbb{R}}} \] The paper considers a generalization of that problem where \(F=[0,1]\) is replaced by a system of intervals \[ [b_1,b_2] \cup \ldots \cup [b_{2p-1},b_{2p}],\quad -\infty <b_1\leq b_2 <\ldots < b_{2p-1}\leq b_{2p} <\infty, \] and where \(\prod_{i=1}^{2n} (1-a_{i,n} x)\) is a real polynomial of degree less than \(2n\) being positive on \([b_1,b_{2p}]\). The solution of the problem is given in a parametric form in terms of automorphic Schottky-Burnside functions.
scientific article; zbMATH DE number 1261155