Applications of the Snake Theorem.

...Keener proves a version of the well-known Snake Theorem of Karlin, and shows the practical use of this theorem in three approximation problems by recasting the problems in the setting of the so-called NL-functional. He then suggests five alternate approximation problems for which he suspects his theorem can be effectively used as an investigative tool. We consider three of these problems: approximation with constraints outside the interval of approximation, reciprocal approximation, and maximin approximation. We show how Theorem 5 of [Keener], Theorem 15 in this thesis, can be effectively used as a tool to apply the snake theorem to them. Approximation with constraints outside the interval of approximation and reciprocal (more specifically rational) approximation were considered in the literature by Laurent and others. We recast these results (or expand and then recast these results) in the setting of the NL-functional and apply Theorem 15 (or a strengthened version of Theorem 15) to obtain the desired oscillation statements. In the case of maximin approximation, to our knowledge, all results are new.