Mergelyan’s theorem

Theorem (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).

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.

References

1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
2 Walter Rudin. . McGraw-Hill, Boston, Massachusetts, 1987.

Title	Mergelyan’s theorem
Canonical name	MergelyansTheorem
Date of creation	2013-03-22 14:23:59
Last modified on	2013-03-22 14:23:59
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30E10
Related topic	RungesTheorem