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.
N. Greene, “A Wavelet-based Method for Overcoming the Gibbs Phenomenon,” in: Recent Advances on Applied Mathematics, Proceedings of the American Conference on Applied Mathematics, Cambridge, Massachusetts, March 24-26, 2008, pp. 408-412.