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 is manifested as spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities.
In this talk we review many of the methods which have been developed in recent years for dealing with this problem. We then describe some of our own contributions and ongoing investigations on the topic, including how wavelets can be used to obtain a multiresolution reconstruction of a function given its Fourier data.