  In a broad sense, positivstellens\"atze are results about representations of
polynomials which are strictly positive on a given set. We give constructive
and, to a large extent, elementary proofs of some known positivstellens\"atze
for compact semialgebraic subsets of $\mathbb{R}^d$. The presented proofs
extend and simplify arguments of Berr, W\"ormann (2001) and Schweighofer (2002,
2005).
