A wavelet-based method for overcoming the Gibbs phenomenon

Authors:
Nataniel Greene
Affiliations:
Department of Mathematics and Computer Science, Kingsborough Community College, CUNY, Brooklyn, NY
Venue:
MATH'08 Proceedings of the American Conference on Applied Mathematics
Year:
2008

Citing 10
Cited 2

On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
An introduction to wavelets

An introduction to wavelets
On the Gibbs Phenomenon and Its Resolution

SIAM Review
Detection of Edges in Spectral Data II. Nonlinear Enhancement

SIAM Journal on Numerical Analysis
Towards the resolution of the Gibbs phenomena

Journal of Computational and Applied Mathematics
On inverse methods for the resolution of the Gibbs phenomenon

Journal of Computational and Applied Mathematics
Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon

Journal of Computational and Applied Mathematics
On the recovery of piecewise smooth functions from their spectral data and integral transforms

On the recovery of piecewise smooth functions from their spectral data and integral transforms
Inverse Polynomial Reconstruction of Two Dimensional Fourier Images

Journal of Scientific Computing
Fourier series of orthogonal polynomials

MATH'08 Proceedings of the American Conference on Applied Mathematics

Fourier series of orthogonal polynomials

MATH'08 Proceedings of the American Conference on Applied Mathematics
Gibbs phenomenon removal by adding Heaviside functions

Advances in Computational Mathematics

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Abstract

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.