Polynomial Selections and Separation by Polynomials

By R, N we denote the set of all reals and positive integers, respectively. Let I ⊂ R be an interval. In this paper we present a necessary and sufficient condition under which two functions f, g : I → R can be separated by a polynomial of degree at most n, where n ∈ N is a fixed number. Our main result is a generalization of the theorem concerning separation by affine functions obtained recently by K. Nikodem and the present author in [3]. To get it we use Behrends and Nikodem’s abstract selection theorem (cf. [1, Theorem 1]). It is a variation of Helly’s theorem (cf. [7, Theorem 6.1]). By cc(R) we denote the family of all non-empty compact real intervals. Recall that if F : I → cc(R) is a set-valued function then a function f : I → R is called a selection of F iff f(x) ∈ F (x) for every x ∈ I . Behrends and Nikodem’s theorem states that if W is an n-dimensional space of functions mapping I into R then a set-valued functionF : I → cc(R) has a selection belonging to W if and only if for every n+ 1 points x1, . . . , xn+1 ∈ I there exists a function f ∈ W such that f(xi) ∈ F (xi) for i = 1, . . . , n + 1. Let us start with the notation used in this paper. Let n ∈ N. If x1, . . . , xn ∈ I are different points then for i = 1, . . . , n we define