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

2001 
Abstract 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 ; A mn ) where A mn 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.
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