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

On regular matrices that induce the Gibbs phenomenon


Author: Joaquin Bustoz
Journal: Proc. Amer. Math. Soc. 25 (1970), 481-487
MSC: Primary 40.31; Secondary 42.00
MathSciNet review: 0259416
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Abstract: Let $ s = \{ {s_n}(z)\} $ be a sequence of complex valued functions defined in a subset $ D$ of the complex plane and suppose that $ {s_n}(z)$ converges to $ f(z)$ for $ z \in D$. For $ {z_0} \in \overline D $ let $ K({z_0};\;s)$ and $ K({z_0};\,f)$ be the cores of $ s$ and $ f$ respectively. We say that $ s$ does not have the Gibbs phenomenon at $ {z_0}$ if $ K({z_0};\,s) \subseteq K({z_0};\,f)$. The regular matrix $ A$ is said to induce the Gibbs phenomenon in $ s$ if $ K({z_0};\,s) \subseteq K({z_0};\,f)$ but $ K({z_0};\,As) \nsubseteq K({z_0};\,f)$. We characterize those regular matrices that induce the Gibbs phenomenon.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0259416-X
PII: S 0002-9939(1970)0259416-X
Keywords: Core, Gibbs phenomenon, regular matrix, Gibbs set
Article copyright: © Copyright 1970 American Mathematical Society