Extensions of endpoint equivalent and periodic Tchebycheff systems (Q1813785)

Let \(A\) be a nonempty set of real numbers with infimum \(\ell_ 1\) and supremum \(\ell_ 2\) which both may be infinite. A real function \(f\) defined on \(A\) is called endpoint equivalent provided that, for all sequences \(s_ k\), \(t_ k\) of points in \(A\) such that \(s_ k\to\ell_ 1\), \(t_ k\to\ell_ 2\) the limits \(f(s_ k)\) and \(f(t_ k)\) exist and are equal. The main result of the paper under review is the theorem: If the set \(A\) contains either \(\ell_ 1\) or \(\ell_ 2\) and for each \(t_ 1<t_ 2\in A\) there is a point \(t_ 3\in A\) with \(t_ 1<t_ 3<t_ 2\), then to any Chebyshev system \(\{x_ 0,x_ 1,...,x_{2n}\}\) on \(A\) consisting of endpoint equivalent functions \(x_ k\) there exist endpoint equivalent functions \(z_ 1\), \(z_ 2\) such that also the extended system \(\{x_ 0,x_ 1,\dots,x_{2n},z_ 1,z_ 2\}\) is a Chebyshev system on \(A\). If the function \(x_ k\) are all continuous, then \(z_ 1\), \(z_ 2\) are also continuous.