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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Polynomial density in Bers spaces
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by Jacob Burbea PDF
Proc. Amer. Math. Soc. 62 (1977), 89-94 Request permission

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 $\left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy$ 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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 89-94
  • MSC: Primary 30A98
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0425139-3
  • MathSciNet review: 0425139