On a theorem of Reiter and spectral synthesis

Motivated by and extending a theorem of Reiter on sets of synthesis in \(\mathbb {R}^N\), we establish a general result for Fourier algebras of locally compact groups, even in the wider context of regular, semisimple and Tauberian commutative Banach algebras, which contains Reiter’s theorem as a special case and explains why it holds. In addition, we give a number of examples and several further results on weak spectral sets.