Sampling theorems provide exact interpolation formulas for bandlimited
functions. They play a fundamental role in signal processing. A function is called
bandlimited if its Fourier transform vanishes outside a compact set. A generalized
sampling theorem in the framework of locally compact Abelian groups is presented.
Sampling sets are finite unions of cosets of closed discrete subgroups. Such sampling
sets are not necessarily periodic and cannot be treated in that setting. An exact reconstruction
formula is found for the case that the support of the Fourier transform
of the function which needs to be reconstructed satisfies certain conditions.
The notion of a filter bank is generalized in the framework of locally compact Abelian
groups. Conditions for perfect reconstruction are derived. It is shown that this
theory includes some generalized sampling theorems and results in multisensor deconvolution
problems as special cases
