THE GIBBS PHENOMENON, THE PINSKY PHENOMENON, AND VARIANTS FOR EIGENFUNCTION EXPANSIONS

Abstract

ABSTRACT We examine analogues of the Gibbs phenomenon for eigenfunction expansions of functions with singularities across a smooth surface, though of a more general nature than a simple jump. The Gibbs phenomena that arise still have a universal form, but a more general class of “fractional sine integrals” arises, and we study these functions. We also make a uniform analysis of eigenfunction expansions in the presence of the Pinsky phenomenon, and see an analogue of the Gibbs phenomenon there. These analyses are done on three classes of manifolds: strongly scattering manifolds, including Euclidean space; compact manifolds without strongly focusing geodesic flows, including flat tori and quotients of hyperbolic space, and compact manifolds with periodic geodesic flow; including spheres and Zoll surfaces. Finally, we uncover a new divergence phenomenon for eigenfunction expansions of characteristic functions of balls, for a perturbation of the Laplace operator on a sphere of dimension ≥5.