  We consider 3D versions of the Zernike polynomials that are commonly used in
2D in optics and lithography. We generalize the 3D Zernike polynomials to
functions that vanish to a prescribed degree $\alpha\geq0$ at the rim of their
supporting ball $\rho\leq1$. The analytic theory of the 3D generalized Zernike
functions is developed, with attention for computational results for their
Fourier transform, Funk and Radon transform, and scaling operations. The
Fourier transform of generalized 3D Zernike functions shows less oscillatory
behaviour and more rapid decay at infinity, compared to the standard case
$\alpha=0$, when the smoothness parameter $\alpha$ is increased beyond 0. The
3D generalized Zernike functions can be used to expand smooth functions,
supported by the unit ball and vanishing at the rim and the origin of the unit
ball, whose radial and angular dependence is separated. Particular instances of
the latter functions (prewavelets) yield, via the Funk transform and the
Fourier transform, an anisotropic function that can be used for a band-limited
line-detecting wavelet transform, appropriate for analysis of 3D medical data
containing elongated structures. We present instances of prewavelets, with
relevant radial functions, that allow analytic computation of Funk and Fourier
transform. A key step here is to identify the special form that is assumed by
the expansion coefficients of a separable function on the unit ball with
respect to generalized 3D Zernike functions. A further issue is how to scale a
function on the unit ball while maintaining its supporting set, and this issue
is solved in a particular form.
