Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Details Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

2010-12-20

arxiv.org/abs/1012.4283v1

Mathematics

Complex Variables

17 pages

Scientific paper

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".

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