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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Another view of the Weierstrass theorem


Author: Kenneth John Preskenis
Journal: Proc. Amer. Math. Soc. 54 (1976), 109-113
MathSciNet review: 0390779
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Abstract: We present two theorems which conclude that polynomials in $ z$ and a given continuous function $ f$ are dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that $ f$ be differentiable and satisfy $ \operatorname{Re} {f_{\overline z }} \geqslant \vert{f_z}\vert$ in the open disk and also that $ {f^{ - 1}}(f(a))$ be countable for each $ a$ in $ D$. The second theorem requires that $ f$ be a class $ {C^1}$-function in a neighborhood of the disk satisfying $ \vert{f_{\overline z }}\vert > \vert{f_z}\vert$ almost everywhere and $ \operatorname{Re} {f_{\overline z }} \geqslant \vert{f_z}\vert$ everywhere inside the disk.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0390779-6
PII: S 0002-9939(1976)0390779-6
Article copyright: © Copyright 1976 American Mathematical Society