  Let L be a Schroedinger operator with periodic magnetic and electric
potentials, a Maxwell operator in a periodic medium, or an arbitrary
self-adjoint elliptic linear partial differential operator in R^n with
coefficients periodic with respect to a lattice G. Let also S be a finite part
of its spectrum separated by gaps from the rest of the spectrum. We consider
the old question of existence of a finite set of exponentially decaying Wannier
functions such that their G-shifts span the whole spectral subspace
corresponding to S in some "nice" manner. It is known that a topological
obstruction might exist to finding exponentially decaying Wannier functions
that form an orthonormal basis of the spectral subspace. This obstruction has
the form of non-triviality of certain finite dimensional (with the dimension
equal to the number of spectral bands in S) analytic vector bundle. We show
that it is always possible to find a finite number of exponentially decaying
composite Wannier functions such that their G-shifts form a 1-tight frame in
the spectral subspace. This appears to be the best, one can do when the
topological obstruction is present. The number of the functions is the smallest
dimension of a trivial bundle containing an equivalent copy of the "obstacle
bundle". In particular, it coincides with the number of spectral bands if and
only if the bundle is trivial, in which case an orthonormal basis of
exponentially decaying composite Wannier functions is known to exist. An
estimate on this number is provided.
