Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Polynomial density in Bers spaces


Author: Jacob Burbea
Journal: Proc. Amer. Math. Soc. 62 (1977), 89-94
MSC: Primary 30A98
MathSciNet review: 0425139
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Abstract: Let D be a bounded Jordan domain such that $ \smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; < \infty $ for $ q > 1$. Here $ {\lambda _D}(z)$ is the Poincaré metric for D. Define $ A_q^p(D)$, the Bers space, to be the Fréchet space of holomorphic functions f on D, such that $ \left\Vert f \right\Vert _{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}\vert f{\vert^p}\;dx\;dy$ is finite, $ 0 < p < \infty ,qp > 1$. It is well known that the polynomials are dense in $ A_q^p(D)$ for $ qp \geqslant 2$. We show that they are dense in $ A_q^p(D)$ for $ qp > 1$ irrespective whether the boundary of D is rectifiable or not.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0425139-3
Keywords: Bers spaces, Poincaré metric, polynomial density
Article copyright: © Copyright 1977 American Mathematical Society