Polynomials that are positive on an interval

This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If $h(x), p(x) \in \mathbb{R}[x]$ such that $\{ \alpha \in \mathbb{R} \mid h(\alpha) \geq 0 \} = [-1,1]$ and $p(x) > 0$ on $[-1,1]$, then there exist sums of squares $s(x), t(x) \in \mathbb{R}[x]$ such that $p(x) = s(x) + t(x) h(x)$. Explicit degree bounds for $s$ and $t$ are given, in terms of the degrees of $p$ and $h$ and the location of the roots of $p$. This is a special case of Schmüdgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval $[0,\infty)$ are also considered.