In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981)[1] and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square of holomorphic functions on a bounded open set in the complex plane by complex polynomials.
It states that complex polynomials form a dense subspace of the Bergman space of a domain bounded by a simple closed Jordan curve.
By the Carathéodory kernel theorem fn(z) converges uniformly on compacta in Ω to z.
[2] In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to z.
On the other hand, by Runge's theorem, hn lies in the closed subspace K of A2(Ω) generated by complex polynomials.