  We present a unified treatment of the Fourier spectra of spherically
symmetric nonlocal diffusion operators. We develop numerical and analytical
results for the class of kernels with weak algebraic singularity as the
distance between source and target tends to $0$. Rapid algorithms are derived
for their Fourier spectra with the computation of each eigenvalue independent
of all others. The algorithms are trivially parallelizable, capable of
leveraging more powerful compute environments, and the accuracy of the
eigenvalues is individually controllable. The algorithms include a Maclaurin
series and a full divergent asymptotic series valid for any $d$ spatial
dimensions. Using Drummond's sequence transformation, we prove linear
complexity recurrence relations for degree-graded sequences of numerators and
denominators in the rational approximations to the divergent asymptotic series.
These relations are important to ensure that the algorithms are efficient, and
also increase the numerical stability compared with the conventional algorithm
with quadratic complexity.
