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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
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