On the recovery of piecewise smooth functions from their spectral data and integral transforms

• Authors: Xiaolin Li;Nataniel Greene
• Venue: On the recovery of piecewise smooth functions from their spectral data and integral transforms
• Year: 2004

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations and integral transform 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. In this dissertation we describe a new theory developed by this author for completely overcoming the Gibbs phenomenon. This new theory allows for the global recovery of a piecewise smooth function without the need for any edge detection. It also applies to approximations in any expansion basis or integral transform which suffers from the Gibbs phenomenon. Also presented in the dissertation are generalizations and contributions to the Gegenbauer reconstruction method of Gottlieb and Shu. These include a new explicit formula for Gegenbauer reconstruction in the general case, a demonstration that Gegenbauer reconstruction can resolve the Gibbs phenomenon in a Fourier-Bessel partial sum, as well as a discussion on the problem of reconstruction for classical orthogonal polynomials apart from the issue of the Gibbs phenomenon, and a Fourier approach to formulating and solving finite difference schemes.