Abstract
A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L 2(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.
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Communicated by John J. Benedetto.
Supported by research grant No 3223 from the Univ. of Crete.
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Kolountzakis, M.N. Periodicity of the Spectrum of a Finite Union of Intervals. J Fourier Anal Appl 18, 21–26 (2012). https://doi.org/10.1007/s00041-011-9187-0
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DOI: https://doi.org/10.1007/s00041-011-9187-0