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Proceedings of the American Mathematical Society

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

 

 

Approximation by polynomials in $ z$ and another function


Author: Kenneth John Preskenis
Journal: Proc. Amer. Math. Soc. 68 (1978), 69-74
MSC: Primary 30A82
DOI: https://doi.org/10.1090/S0002-9939-1978-0457740-6
MathSciNet review: 0457740
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Abstract: We present some progress in the understanding of when and why polynomials in z and a given continuous function f are uniformly dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that f be an $ {\text{ACL}^2}$-function in a neighborhood of the disk which satisfies $ \operatorname{Re} {f_{\bar z}} \geqslant \vert{f_z}\vert$ almost everywhere and $ {f^{ - 1}}(f(a))$ is countable for each a. The second theorem requires that f have a special form and satisfy $ \vert{f_{\bar z}}\vert > \vert{f_z}\vert$ everywhere except at the origin. The form is that $ f = {\bar z^k}\phi (\vert z{\vert^{2k}})$ where $ \phi $ is a complex valued function of a real variable satisfying $ \phi $ is continuous in [0, 1], $ \phi '$ exists in (0, 1) and where k is a positive integer.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0457740-6
Article copyright: © Copyright 1978 American Mathematical Society