  Gabor frames with Hermite functions are equivalent to sampling sequences in
true Fock spaces of polyanalytic functions. In the L^2-case, such an
equivalence follows from the unitarity of the polyanalytic Bargmann transform.
We will introduce Banach spaces of polyanalytic functions and investigate the
mapping properties of the polyanalytic Bargmann transform on modulation spaces.
By applying the theory of coorbit spaces and localized frames to the Fock
representation of the Heisenberg group, we derive explicit polyanalytic
sampling theorems which can be seen as a polyanalytic version of the lattice
sampling theorem discussed by J. M. Whittaker in Chapter 5 of his book
"Interpolatory Function Theory".
