Smooth interpolation of lattice gauge fields by signal processing methods

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 ($C^{\mathrm{\infty }}$), 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 $A_{\mu }^c(x)$. We then discuss the use of a maximum entropy type method to address gauge invariance in the non-abelian case.