The snake theorem for unisolvent families (Q2366708)

One of the most beautiful theorems of analysis is the oscillation theorem due to \textit{S. Karlin} [J. Math. Mech. 12, 599-617 (1963; Zbl 0132.048)], termed as `Snake Theorem'. The theorem shows under appropriate conditions the existence of a function \(p^*\) from a Chebyshev space \(T\), with a graph that alternately `touches' the graphs of functions \(f\) and \(g\) where \(f<g\) and \(f\leq p^*\leq g\) on a compact interval \([a,b]\). The number of `touches' depends upon the dimension of \(T\) (Two functions \(u\) and \(v\) defined on \([a,b]\) are said to touch at \(x_ 0\) in \([a,b]\) if there is a sequence \((x_ i)\) in \([a,b]\) such that \(x_ i\to x_ 0\) and \(u(x_ i)-v(x_ i)\to 0)\). A number of alternate proofs of this theorem and various extensions of this theorem have appeared in the literature. In this paper it is shown that the theorem is valid for (almost) arbitrary unisolvent families where there are no continuity requirements on the upper and lower bounding functions. It is also shown that under an essential additional hypothesis, the theorem is also true for unisolvent families. A number of applications to uniform approximation from unisolvent families and Chebyshev spaces are considered in the paper.