Farrell–Markushevich theorem

In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) 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. The Gram–Schmidt process can be used to construct an orthonormal basis in the Bergman space and hence an explicit form of the Bergman kernel, which in turn yields an explicit Riemann mapping function for the domain.

Proof

Let Ω be the bounded Jordan domain and let Ωn be bounded Jordan domains decreasing to Ω, with Ωn containing the closure of Ωn + 1. By the Riemann mapping theorem there is a conformal mapping fn of Ωn onto Ω, normalised to fix a given point in Ω with positive derivative there. By the Carathéodory kernel theorem fn(z) converges uniformly on compacta in Ω to z.[1] In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to z. Given a subsequence of fn, it has a subsequence, convergent on compacta in Ω. Since the inverse functions converge to z, it follows that the subsequence converges to z on compacta. Hence fn converges to z on compacta in Ω.

As a consequence the derivative of fn tends to 1 uniformly on compacta.

Let g be a square integrable holomorphic function on Ω, i.e. an element of the Bergman space A2(Ω). Define gn on Ωn by gn(z) = g(fn(z))fn'(z). By change of variable

${\displaystyle \displaystyle {\|g_{n}\|_{\Omega _{n}}^{2}=\|g\|_{\Omega }^{2}.}}$

Let hn be the restriction of gn to Ω. Then the norm of hn is less than that of gn. Thus these norms are uniformly bounded. Passing to a subsequence if necessary, it can therefore be assumed that hn has a weak limit in A2(Ω). On the other hand hn tends uniformly on compacta to g. Since the evaluation maps are continuous linear functions on A2(Ω), g is the weak limit of hn. On the other hand, by Runge's theorem, hn lies in the closed subspace K of A2(Ω) generated by complex polynomials. Hence g lies in the weak closure of K, which is K itself.[2]