We show how to set up a Cellular Dynamical Mean Field Theory for
the Holstein's polaron problem using the exact solution of a
cluster of n sites embedded in a Weiss's field. We show that a
restricted basis, that allows excitations of phonons only for n
sites at a time, reproduces exactly the equations of the n-site
Dynamical Mean Field Theory, and enables to check the proposed
decoupling scheme of the Green's functions via Exact Numerical
Diagonalizations. We introduce a real space formulation of the
Cellular Dynamical Mean Field Theory that applies to any lattice
with or without periodic boundary conditions and that allows to
partition the lattice into different kinds of clusters.
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