Abstract
Throughout this note, G = {x} denotes a compact Abelian group and X = {x } its character group. We first exhibit a reciprocal relationship between generalised versions of the Parseval formula and nets (T.) of endomorphisms of L 1 G) which converge in a suitable sense to the identity endomorphism, and which are such that, for each f ϵ L 1 (G), T i f has an absolutely convergent Fourier series and is approximable arbitrarily closely by linear combinations of translates of f. In this way formulae of the Parseval type are seen to have a genesis in the approximation problems of harmonic analysis and synthesis.
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