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 | References | Similar Articles | Additional Information
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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- Richard G. Cooke, Infinite Matrices and Sequence Spaces, Macmillan & Co., Ltd., London, 1950. MR 0040451 W. A. Hurwitz, Some properties of methods of evaluation of divergent sequences. Proc. London Math. Soc. (2) 26 (1927), 231-248.
- Shin-ichi Izumi and Masako Satô, Fourier series. X. Rogosinski’s lemma, K\B{o}dai Math. Sem. Rep. 8 (1956), 164–180. MR 86179
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Keywords:
Core,
Gibbs phenomenon,
regular matrix,
Gibbs set
Article copyright:
© Copyright 1970
American Mathematical Society