  We propose, analyze, and implement interpolatory approximations and
Filon-type cubature for efficient and accurate evaluation of a class of
wideband generalized Fourier integrals on the sphere. The analysis includes
derivation of (i) optimal order Sobolev norm error estimates for an explicit
discrete Fourier transform type interpolatory approximation of spherical
functions; and (ii) a wavenumber explicit error estimate of the order
$\mathcal{O}(\kappa^{-\ell} N^{-r_\ell})$, for $\ell = 0, 1, 2$, where $\kappa$
is the wavenumber, $N$ is the number of interpolation/cubature points on the
sphere and $r_\ell$ depends on the smoothness of the integrand. Consequently,
the cubature is robust for wideband (from very low to very high) frequencies
and very efficient for highly-oscillatory integrals because the quality of the
high-order approximation (with respect to quadrature points) is further
improved as the wavenumber increases. This property is a marked advantage
compared to standard cubature that require at least ten points per wavelength
per dimension and methods for which asymptotic convergence is known only with
respect to the wavenumber subject to stable of computation of quadrature
weights. Numerical results in this article demonstrate the optimal order
accuracy of the interpolatory approximations and the wideband cubature.
