Speed of convergence of two-dimensional Fourier integrals

1. Introduction Recently [2,3] we found necessary and sufficient conditions for the convergence at a preassigned point of the spherical partial sums of the Fourier integral in a class of piece-wise smooth functions in Euclidean space. These yield elementary examples of divergent Fourier integrals in three dimensions and higher. Meanwhile, several years ago Gottlieb and Orsag[1] observed that in two dimensions we may expect slower convergence at certain points, specifically for Fourier-Bessel series of radial functions. In this paper we investigate the rate of convergence of the spherical partial sums of the Fourier integral for a class of piecewise smooth functions. The basic result is an asymptotic expansion which allows us to read off the rate of convergence at a pre-assigned point.