Abstract Harmonic Analysis and the Sampling Theorem*

Abstract

Abstract The step from Fourier analysis to abstract harmonic analysis is a natural one which unites a number of apparently disparate results into one general framework with a concise and elegant notation. The classical sampling theorem is a good example of this, since a number of different versions can be obtained from the one abstract result due to I. Kluvanek (1965) simply by choosing the appropriate underlying locally compact abelian group. J.R. Higgins (1985) gives several examples where the sampling theorem is obtained for the real line ℝ, the circle 𝕋, the torus 𝕋, the integers ℤ, the dyadic group and Euclidean space ℝ n,. Haar measure, a translation-invariant measure associated with the locally compact abelian group, plays a prominent role in the abstract theory; reciprocity relations and measure relations between the Haar measures of certain groups and sets emerge as natural and useful. The coset decomposition formula (10.2.3) below plays a particularly important part as it allows the analysis to be carried out in the space of square-integrable functions on a compact abelian group.