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 \({{\mathcal {M}}}(G)\).
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Communicated by Hans G. Feichtinger.
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Platonov, S.S. On Spectral Analysis and Spectral Synthesis in the Space of Tempered Functions on Discrete Abelian Groups. J Fourier Anal Appl 24, 1340–1376 (2018). https://doi.org/10.1007/s00041-017-9567-1
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DOI: https://doi.org/10.1007/s00041-017-9567-1
Keywords
- Spectral synthesis
- Spectral analysis
- Locally compact Abelian groups
- Tempered functions
- Bruhat–Schwartz functions