Polynomial Runge theorem

If $K\subset \mathbb{C}$ is compact with connected complement and $f$ is holomorphic near $K$, then for each $\epsilon >0$ there is a polynomial $p$ such that ${\sup }_K|p-f|<\epsilon$.

Let the uniform closure of polynomials on a compact set be an algebra. A reciprocal at a sufficiently distant pole belongs to it by a geometric series. A path of poles avoiding the compact set can be divided into short steps; expanding each reciprocal in powers of the previous one transfers polynomial approximability along the path.

For a closed set $K⊊\mathbb{C}$, a point $\lambda \notin K$ is a polynomially approximable resolvent point when $z\mapsto (z-\lambda {)}^{-1}$ is a uniform limit on $K$ of complex polynomials.

The polynomially approximable resolvent points form a subset that is both open and closed relative to $\mathbb{C}\setminus K$. Openness follows from the uniformly convergent expansionwhen $|\mu -\lambda |<\mathrm{dist}(\lambda ,K)$; relative closedness follows from the uniform continuity of the resolvent as its pole varies away from $K$. Hence approximability at one point propagates throughout its connected component of $\mathbb{C}\setminus K$.

Let $S$ be a closed square and let $f$ be holomorphic on a neighbourhood of a slightly larger square. The Cauchy integral formula on the larger boundary is uniformly approximated on $S$ by Riemann sums of resolvents whose poles lie outside $S$. Far-away resolvents have convergent geometric-series expansions in $z$, and resolvent propagation across a complementary component moves this polynomial approximability to every pole outside the square. Approximating the finitely many resolvents in each Riemann sum by polynomials proves polynomial approximation on $S$ without invoking the general Runge theorem.

A proper closed arc $S$ of a circle centered at zero has connected complement and stays away from zero. The polynomial Runge theorem therefore gives polynomials converging uniformly to $1/z$ on $S$.

On $K=\{z:|z|=1,\mathrm{Re}z\le 0\}$, the polynomialsconverge uniformly to $1/z$. Expand $1/z$ first in powers of $4/(z-4)$, whose modulus is at most $4/\sqrt{17}$ on $K$, and then expand each $(z-4{)}^{-j-1}$ in powers of $z/4$. The estimate $\left(\genfrac{}{}{0ex}{}{j+k}{k}\right)\le 2^{j+k}$ controls the diagonal truncation.

Polynomials cannot converge uniformly to $1/z$ on $\{z:|z|=1,z\ne 1\}$. Uniform convergence there makes the polynomials uniformly Cauchy on the full circle because deleting one point does not change the supremum of a continuous function. They would therefore converge uniformly to $1/z$ on the full circle, contradicting

To approximate a piecewise constant function pointwise, exhaust each of its separated regions by compact subsets whose finite union has connected complement. Apply polynomial Runge approximation to the locally constant holomorphic function on each stage, with errors tending to zero. Every fixed point eventually lies in all later stages, so the uniform stagewise estimates imply pointwise convergence.