The phenomenon involves a lack of convergence at a point due to a discontinuity at boundary.
The jump at |x| = c will cause an oscillatory behavior of the spherical partial sums, which prevents convergence at the center of the ball as well as the possibility of Fourier inversion at x = 0.
Stated differently, spherical partial sums of a Fourier integral of the indicator function of a ball are divergent at the center of the ball but convergent elsewhere to the desired indicator function.
Also related are eigenfunction expansions on a geodesic ball in a rank-one symmetric space, but one must consider boundary conditions.
Pinsky and others also represent some results on the asymptotic behavior of the Fejer approximation in one dimension, inspired by work of Bump, Persi Diaconis, and J. B.