  We digitally filter the Fourier modes of the link angles of an abelian
lattice gauge field which produces the Fourier modes of a continuum $A_\mu(x)$
that exactly reproduces the lattice links through their definition as phases of
finite parallel transport. The constructed interpolation is smooth
($C^\infty$), free from transition functions, and gauge equivariant. After
discussing some properties of this interpolation, we discuss the non-abelian
generalization of the method, arriving for SU(2), at a Cayley parametrization
of the links in terms of the Fourier modes of $A^c_\mu(x)$. We then discuss the
use of a maximum entropy type method to address gauge invariance in the
non-abelian case.

High Energy Physics - Lattice