Rational and Polynomial Density on Compact Real Manifolds

Abstract

We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n^{\prime }$ functions $g_1,...,g_{n^{\prime }}$ in ${\mathcal{C}}^{\infty }(M)$ so that every element of ${\mathcal{C}}^k(M)$ can be approximated by rational combinations of $f_1,...,f_n$ and polynomial combinations of $g_1,...,g_{n^{\prime }}$. As an application, we show that the optimal value of $n$ and $n^{\prime }$ for all manifolds of dimension $m$ is [3m/2], when $k\ge 1$ and $m\ge 2$.

arXiv comment17 Pages; definitions have been slightly modified