Convolution equations in many-dimensional domains and on the Heisenberg reduced group

Local versions of the Brown-Schreiber-Taylor theorem on spectral analysis in $\mathbb R^n$ are obtained under most general assumptions. This has made it possible, in particular, to prove the equivalence of the global and the local Pompeiu properties for a compact subset $E$ of $\mathbb R^n$ without any assumptions on $E$. Perfect analogues of these results are established for systems of convolution equations on the Heisenberg group $H^n_{\mathrm{red}}$. As an application, for subspaces of $C(H^n_{\mathrm{red}})$ invariant under shifts and unitary transformations a spectral synthesis theorem is proved, analogues of which were known before only for functions of slow growth.
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