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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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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0259416-X
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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Keywords: Core, Gibbs phenomenon, regular matrix, Gibbs set
Article copyright: © Copyright 1970 American Mathematical Society