The Gibbs phenomenon for piecewise-linear approximation

@article{Foster1991TheGP,
  title={The Gibbs phenomenon for piecewise-linear approximation},
  author={J. Foster and Franklin Richards},
  journal={American Mathematical Monthly},
  year={1991},
  volume={98},
  pages={47-49}
}
In 1899 J. W. Gibbs [1] of Yale, in response to a letter in Nature by the American physicist A. Michelson [2], presented a result about Fourier series that now goes by the name of the Gibbs phenomenon. Michelson had complained about an undesired "overshoot" effect that occurred whenever he approximated a function having a jump discontinuity by a finite Fourier series. Gibbs was able to show that this overshoot does not disappear as the number of terms in the series becomes arbitrarily large… 

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I SHOULD like to add a few words concerning the subject of Prof. Michelson's letter in NATURE of October 6. In the only reply which I have seen (NATURE, October 13), the point of view of Prof.
Fourier's Series
IN all expositions of Fourier's series which have come to my notice, it is expressly stated that the series can represent a discontinuous function.
Undercurrents in the Strait of Bab-el-Mandeb
AN interesting observation has recently been made by one of H.M. surveying vessels, and I forward the Preface to the account of the details published by the Hydrographic Department, which contains
1899) 606. (This letter corrected an error he had made in a letter of
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  • 1898