# 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…

## 61 Citations

The Gibbs phenomenon for bestL1-trigonometric polynomial approximation

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- 1995

The classical Gibbs phenomenon for the Fourier sections (bestL2-trigonometric polynomial approximants) of a jump function asserts that, near the jump, these sections “overshoot” the function by an…

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By providing a sufficient condition to compute the size of Gibbs phenomenon for the Shannon wavelet series, the overshoot is propotional to the jump at discontinuity and by comparing it with that of the Fourier series, it is seen that these two have exactly the same Gibbs constant.

GIBBS PHENOMENON FOR TRIGONOMETRIC INTERPOLATION

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The Gibbs’ phenomenon for the classical Fourier series is known. This occurs for almost all series expansions. This phenomenon has been observed even in sampling series.In this paper, we show the…

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Nonlinear state-space models driven by differential equations have been widely used in science. Their statistical inference generally requires computing the mean and covariance matrix of some…

Bi-orthogonal wavelets for investigating Gibbs effects via oblique extension principle

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Gibbs effect is generally known for Fourier and Wavelets expansions of a function in the neighborhood of its discontinuities points which deals with the nonuniform convergence of its truncated sums…

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This thesis unifies two optimal diffusion stopping principles and suggests that the outcome of the diffusion should be interpreted as a certain expectation conditioned on the initial signal of observations instead of being treated as a random sample or probabilities.

The Fourier-series method for inverting transforms of probability distributions

- MathematicsQueueing Syst. Theory Appl.
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This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions and describes two methods for inverting Laplace transform based on the Post-Widder inversion formula.

Persistent Homology

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Consider the art of Seurat or a piece of old newsprint. The eye, or the brain, performs the marvelous task of taking the sense data of individual points and assembling them into a coherent image of a…

Gibbs Phenomenon Resolution in 2&3 Dimensional Space via Mupad Micro Models

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This paper aims to reconstruct Mupad models to treat & resolve the overshoots & tautologically undershoots that usually appears while approximating continuous or piecewise continuous function through…

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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

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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

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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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