ON THE GIBBS PHENOMENON AND ITS RESOLUTION

The nonuniform convergence of the Fourier series for discontinuous fun ctions, and in particular the oscillatory behavior of the finite sum. was already analyzed by Wilbraham in 1848. This was later named the Gi bbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals wi th the issue of recovering point values of a function from its expansi on coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main th eme here is not the structure of the Gibbs oscillations but the unders tanding and resolution of the phenomenon in a general setting. The pur pose of this article is to review the Gibbs phenomenon and to show tha t the knowledge of the expansion coefficients is sufficient for obtain ing the point values of a piecewise smooth function, with the same ord er of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approx imation.