  By using the overcompleteness of coherent states we find an alternative form
of the unit operator for which the ket and the bra appearing under the
integration sign do not refer to the same phase-space point. This defines a new
quantum representation in terms of Bargmann functions, whose basic features are
presented. A continuous family of secondary reproducing kernels for the
Bargmann functions is obtained, showing that this quantity is not necessarily
unique for representations based on overcomplete sets. We illustrate the
applicability of the presented results by deriving a semiclassical expression
for the Feynman propagator that generalizes the well-known van Vleck formula
and seems to point a way to cope with long-standing problems in semiclassical
propagation of localized states.
