Approximating signals in the abstract

Abstract

An analogue of the classical approximate sampling theorem is proved for the abstract analogue of a signal, i.e. a function on a locally compact abelian group that is continuous, square-integrable with an integrable Fourier–Plancherel transform. An additional hypothesis that the samples of the function are square-summable is needed and is discussed. This hypothesis is not very restrictive as in a sense it ‘almost always’ holds. Two asymptotic formulae are also obtained under some further conditions on the group.