Ramanathan J., Steger T.
Incompleteness of Sparse Coherent States
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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.