Polynomial snakes with respect to subsystems of algebraic degrees (Q1589021)

Let be given a fixed \(n\in \mathbb{N}\) and a pair \((g,G)\) of functions defined on the interval \([a,b]\), the set \(L(g,G)\) is defined by \[ L(g,G)=\{p\in\Pi_n: g(x)\leq p(x)\leq G(x)\text{ for all } x\in [a,b]\}. \] Now, a polynomial \(p\in L(g,G)\) is called a snake with respect to the system of functions \((g,G)\) if there exists a polynomial \(p\in L(g,G)\) and a set of \(m=n+1\) points \(\{x_0<x_1<\dots <x_n\}\subset [a,b]\) such that, with \(k\) growing from \(1\) to \(m\), the points \(M_k=(x_k,p(x_k))\) jump from the graph of G to the graph of g alternating. This concept, which is clearly connected to the Chebyshev alternance theorem, was introduced by Karling in 1963 and has been studied by several Russian mathematicians (Dzyadyk, Dolzenko, Sebastyanov and others). In the paper the author studies some existence results for certain Chebychev-type systems of functions.