Mergelyan's theoremMergelyansTheorem2004-06-07 14:42:502007-12-04 19:02:30Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{thm}[Mergelyan]
Let $K \subset {\mathbb{C}}$ be a compact subset of the complex plane such that
${\mathbb{C}} \backslash K$ (the complement of $K$) is connected, and let
$f\colon K \to {\mathbb{C}}$ be a continuous function which is also holomorphic
on the interior of $K.$ Then $f$ is the uniform limit on $K$ of holomorphic
polynomials (polynomials in one complex variable).
\end{thm}
So for any $\epsilon > 0$ one can find a polynomial $p(z) = \sum_{j=1}^n a_j z^j$
such that $\lvert f(z) - p(z) \rvert < \epsilon$ for all $z \in K.$
Do note that this theorem is not a weaker version of Runge's theorem. Here, we do not
need $f$ to be holomorphic on a neighbourhood of $K,$ but just on the interior of $K.$ For example, if the interior of $K$ is empty, then $f$ just needs to be continuous on $K.$ Further, it could be that the closure of the interior of $K$
might not be all of $K.$ Consider $K = D \cup [-10,10],$ where $D$
is the closed unit disc. Then $K$ has two lines coming out of either end of the disc and $f$ needs to only be continuous there.
Also note that this theorem is distinct from the Stone-Weierstrass theorem. The point here is that the polynomials are
holomorphic in Mergelyan's theorem.
\begin{thebibliography}{9}
\bibitem{Conway:complexI}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable I}}.
Springer-Verlag, New York, New York, 1978.
\bibitem{Rudin:realcomplex}
Walter Rudin.
{\em \PMlinkescapetext{Real and Complex Analysis}}.
McGraw-Hill, Boston, Massachusetts, 1987.
\end{thebibliography}