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Transactions of the American Mathematical Society

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Weighted $ L\sp{2}$ approximation of entire functions


Author: Devora Wohlgelernter
Journal: Trans. Amer. Math. Soc. 202 (1975), 211-219
MSC: Primary 30A82
MathSciNet review: 0355069
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Abstract: Let $ S$ be the space of entire functions $ f(z)$ such that $ \vert\vert f(z)\vert{\vert^2} = \smallint \smallint \vert f(z){\vert^2}dm(z)$, where $ m$ is a positive measure defined on the Borel sets of the complex plane. Write $ dm(z) = K(z)d{A_z} = K(r,\theta )dAz$. Theorem 1. If $ \ln {\inf _\theta }K(r,\theta )$ is asymptotic to $ \ln {\sup_\theta} K(r,\theta )$ (together with other mild restrictions) then polynomials are dense in $ S$. Theorem 2. Let $ K(z) = {e^{ - \phi (z)}}$ where $ \phi (z)$ is a convex function of $ z$ such that all exponentials belong to $ S$. Then polynomials are dense in $ S$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0355069-X
Keywords: Weighted $ {L^2}$ approximation, complete, dense
Article copyright: © Copyright 1975 American Mathematical Society