Spectrality of product domains and Fuglede’s conjecture for convex polytopes

Rachel Greenfeld, Nir Lev

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6 Scopus citations


A set Ω ⊂ ℝ2 is said to be spectral if the space L2(Ω) has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets “behave like” sets which can tile the space by translations. This suggests a conjecture that a product set Ω = A × B is spectral if and only if the factors A and B are both spectral sets. We recently proved this in the case when A is an interval in dimension one. The main result of the present paper is that the conjecture is true also when A is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope Ω is spectral if and only if it can tile by translations.

Original languageEnglish
Pages (from-to)409-441
Number of pages33
JournalJournal d'Analyse Mathematique
Issue number2
StatePublished - 1 Mar 2020

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© 2020, The Hebrew University of Jerusalem.


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