Abstract
We consider some problems of spectral analysis and spectral synthesis in the topological vector space $${{\mathcal {M}}}(G)$$ of tempered functions on a discrete Abelian group G. It is proved that spectral analysis holds in the space $${{\mathcal {M}}}(G)$$ on every Abelian group G, that is, every nonzero closed linear translation invariant subspace of $${{\mathcal {M}}}(G)$$ contains an exponential. For any finitely generated Abelian group G it is proved, that spectral synthesis holds in $${{\mathcal {M}}}(G)$$ , that is, every closed linear translation invariant subspace $${{\mathscr {H}}}$$ of $${{\mathcal {M}}}(G)$$ coincides with the closed linear span of all exponential monomials belonging to $${{\mathscr {H}}}$$ . For any Abelian group G with infinite torsion free rank it is proved that spectral synthesis fails to hold in the space $${{\mathcal {M}}}(G)$$ .
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have