Optimal bases of Gaussians in a Hilbert space: applications in mathematical signal analysis

Arbitrary square-integrable (normalized) functions can be expanded exactly in terms of the Gaussian basis g(t;A) where A∈C. Smaller subsets of this highly overcomplete basis can be found, which are also overcomplete, e.g., the von Neumann lattice g(t;Amn) where Amn are on a lattice in the complex plane. Approximate representations of signals, using a truncated von Neumann lattice of only a few Gaussians, are considered. The error is quantified using various p-norms as accuracy measures, which reflect different practical needs. Optimization techniques are used to find optimal coefficients and to further reduce the size of the basis, whilst still preserving a good degree of accuracy. Examples are presented.