Treffer: Nonperiodic sampling theorems and filter banks

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.