An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.Received February 7, 2002; in revised form May 24, 2002Published online April 4, 2003