*[It arises from consideration of where the weight is exponentially large. -T.]*

There are various ways to establish Plancherel’s theorem (which includes the Fourier inversion formula as a component) on arbitrary locally compact abelian (LCA) groups; see Section 4 of these notes for a brief discussion of two approaches, one using Gelfand theory, and another via Bochner’s theorem. For special groups such as the real line or the circle there are shortcuts available that utilise special features of those groups, such as the presence of gaussians or holomorphic extensions to various regions of the complex plane, but these do not seem to generalise well to the LCA setting, which seems to be the most natural framework in which to present Fourier analysis in a unified manner.

]]>We also know that for the discrete setting we can give a proof of the Fourier inversion formula which uses some algebraic manipulations and orthogonality properties.

In general is there a unified approach to prove Fourier inversion type results?

]]>*[Corrected, thanks – T.]*

*[Here we use to denote the estimate (that is, the absolute values are implicit in the notation) -T.]*

*[As it turns out, one does not need a dependence on here. -T]*