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Spectrum is periodic for \(n\)-intervals. (English) Zbl 1253.42028

Summary: We study spectral sets which are unions of finitely many intervals in \(\mathbb R\). We show that any spectrum associated with such a spectral set \(\varOmega \) is periodic, with the period an integral multiple of the measure of \(\varOmega \). As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.

MSC:

42C15 General harmonic expansions, frames
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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