Farrell–Markushevich theorem

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 f**n of Ωn onto Ω, normalised to fix a given point in Ω with positive derivative there. By the Carathéodory kernel theorem f**n(z) converges uniformly on compacta in Ω to z.See:

In fact Carathéodory's theorem implies that the inverse maps tend uniformly on compacta to z. Given a subsequence of f**n, 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 f**n converges to z on compacta in Ω.

As a consequence the derivative of f**n 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 g**n on Ωn by g**n(z) = g(f**n(z))f**n'(z). By change of variable

:\displaystyle{|g_n|^2_{\Omega_n} =|g|_\Omega^2.}

Let h**n be the restriction of g**n to Ω. Then the norm of h**n is less than that of g**n. Thus these norms are uniformly bounded. Passing to a subsequence if necessary, it can therefore be assumed that h**n has a weak limit in A2(Ω). On the other hand, h**n tends uniformly on compacta to g. Since the evaluation maps are continuous linear functions on A2(Ω), g is the weak limit of h**n. On the other hand, by Runge's theorem, h**n 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.

Notes

References

• {{citation|last=Farrell|first= O. J.|title= On approximation to an analytic function by polynomials|journal= Bull. Amer. Math. Soc. |volume=40|year=1934|issue= 12|pages=908–914|doi=10.1090/s0002-9904-1934-06002-6|doi-access=free}}

References

1. Bick, Theodore A.. (9 December 2019). ["A History of the Mathematics Department"](https://www.union.edu/mathematics/history-mathematics-department}}; {{cite web).
2. {{harvnb. Conway. 2000