en>Apoptosa: /* la (na) aligned with null radial geodesics */

2013-05-29T23:07:01Z

la (na) aligned with null radial geodesics

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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 function]]s 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 theorem|Riemann mapping function]] for the domain.

==Proof==
Let Ω be the bounded Jordan domain and let Ω''n'' be bounded Jordan domains decreasing to Ω, with Ω''n'' containing the closure of Ω''n'' + 1. By the Riemann mapping theorem there is a conformal mapping ''f''''n'' of Ω''n'' 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''.<ref>See:
*{{harvnb|Conway|2000|pp=150–151}}
*{{harvnb|Markushevich|1967|pp=31–35}}
</ref> 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 Ω''n'' by ''g''''n''(''z'') = ''g''(''f''''n''(''z''))''f''''n'''(''z''). By change of variable

:<math>\displaystyle{\|g_n\|^2_{\Omega_n} =\|g\|_\Omega^2.}</math>

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 ''A''2(Ω) generated by complex polynomials. Hence ''g'' lies in the weak closure of ''K'', which is ''K'' itself.<ref>{{harvnb|Conway|2000|pp=151–152}}</ref>

==See also==
* [[Mergelyan's theorem]]

==Notes==
{{reflist}}

==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|pages=908–914}}
*{{citation|last=Markushevich|first= A. I.|title= Theory of functions of a complex variable. Vol. III|publisher=Prentice–Hall|year=1967}}
*{{citation| last=Conway|first= John B.|title= A course in operator theory|series= Graduate Studies in Mathematics|volume= 21|publisher=American Mathematical Society|year= 2000|isbn= 0-8218-2065-6}}

{{DEFAULTSORT:Farrell-Markushevich theorem}}
[[Category:Mathematical theorems]]
[[Category:Operator theory]]
[[Category:Functional analysis]]
[[Category:Complex analysis]]

Construction of a complex null tetrad - Revision history

en>Apoptosa: /* la (na) aligned with null radial geodesics */