Fourier series of radial functions in several variables

We prove that the spherical partial sums of the Fourier series of the indicator function of a ball inside the cube of width 2π converge at the center of the ball if and only if the dimension is strictly less than three. For more general radial functions in three dimensions we give a necessary and sufficient condition for convergence. Related questions of non-localization and convergence of Fourier transforms of radial functions in three dimensions are examined.