Oka–Weil theorem

Remarks

The proof first finds polynomials $p_1, \dots, p_N$ such that the polynomial polyhedron $P = \set{z : \abs{z_j} \le R, \ \abs{p_k(z)} \le 1}$ contains $K$ and lies in the open set where $f$ is defined; this uses polynomial convexity of $K$. The map $z \mapsto (z, p_1(z), \dots, p_N(z))$ embeds a neighbourhood of $P$ as a closed complex submanifold of a polydisc in $\C^{n+N}$. Oka's extension theorem extends $f$ from this submanifold to a holomorphic function on the polydisc, and the Taylor series of the extension gives polynomials converging to $f$ uniformly on $K$.

Example: for $n = 1$ the theorem is the polynomial case of Runge's theorem: if $\C \setminus K$ is connected, every function holomorphic near $K$ is a uniform limit of polynomials on $K$.

Polynomial convexity cannot be dropped. On the unit circle $K$, which is not polynomially convex, the function $1/z$ is not a uniform limit of polynomials: polynomials $p$ satisfy $\int_K p(z) \, dz = 0$, while $\int_K z^{-1} \, dz = 2\pi i$.

Going deeper

For a student

The theorem is a basic approximation result of several complex variables. It generalises to holomorphically convex compact subsets of Stein manifolds and of domains of holomorphy, with polynomials replaced by holomorphic functions on the whole domain. The method of its proof, embedding a polynomial polyhedron into a polydisc, reduces global questions to questions on polydiscs. It is also used to show that convex domains and balls are Runge domains.