On the Fourier transform on function algebras on locally compact Abelian groups

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We obtain a characterisation of the Fourier transform on the space of Schwartz–Bruhat functions and the Feichtinger algebra on locally compact Abelian groups. The result states that any appropriately additive bijection from the Schwartz–Bruhat space (resp. the Feichtinger algebra) of a locally compact Abelian group onto the Schwartz–Bruhat space (resp. the Feichtinger algebra) of its dual, which interchanges convolution and pointwise products, is essentially the Fourier transform. We obtain this as a consequence of a similar result for the more general class of algebras, which we call Fourier twin algebras.