Periodicity of the spectrum in dimension one

2011/08/29 by Iosevich, Alex, Kolountzakis, Mihail N. · 4 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1108.5689

Abstract

A bounded measurable set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers ("frequencies") such that the exponential functions e_λ(x) = exp(2πi λx), λ∈Λ, form a complete orthonormal system of L²(Ω). Such a set Λ is called a \em spectrum of Ω. In this note we prove that any spectrum Λ of a bounded measurable set Ω⊆\RR must be periodic.

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