94-95 Ramanathan J., Steger T.

Incompleteness of Sparse Coherent States (25K, LaTeX) Apr 18, 94

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Abstract. We study necessary conditions for a set of phase space translates of an arbitrary function $f \in L^2({\bf R}^n)$ to be complete, a frame or a Riesz basis. The necessary conditions are given in terms of the density of the underlying set of phase space translates. Among our results is an elementary proof of a theorem of M. Rieffel's that states that the set of phase space translates of a function is always incomplete if the underlying set is a lattice of density less than one. We also prove that if a set of uniformly discrete phase space translates of a square integrable function forms a Riesz basis, then the asymptotic density must be exactly equal to one. An improvement of a result of H.J. Landau concerning necessary density conditions for a set of coherent states to be a frame is also given.

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