Shannon-type sampling theory on unions of equally spaced and non-commensurate grids

Sampling Theory is that branch of mathematics which seeks to reconstruct functions from their values at a discrete set of points. The fundamental result in Sampling Theory, known as Shannon's Sampling Theorem, has many applications to signal processing and communications engineering. We will demonstrate Shannon's result via complex interpolation methods. We then quote a result of Casey which uses these methods to solve interpolation problems on unions of non-commensurate lattices, which are created via specific number theoretic guidelines. These interpolations give Shannon-type reconstructions on these lattices. We close by doing simulations in MATLAB of the sampling reconstructions on these non-commensurate grids.