Rational and Polynomial Density on Compact Real Manifolds

TL;DR

This paper characterizes when smooth functions on compact real manifolds can be approximated by rational and polynomial combinations of a finite set of functions, providing optimal bounds for the number of such functions needed.

Contribution

It offers a new characterization for the approximation of smooth functions on manifolds using rational and polynomial functions, including optimal bounds for the number of functions required.

Findings

Established a characterization for approximation by rational and polynomial functions.

Determined the optimal number of functions needed for approximation on manifolds.

Provided bounds of [3m/2] for the number of functions based on manifold dimension.

Abstract

We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n'$ functions $g_1,...,g_{n'}$ 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 $g_1,...,g_{n'}$. As an application, we show that the optimal value of $n$ and $n'$ for all manifolds of dimension $m$ is [3m/2], when $k\geq 1$ and $m\geq 2$.