Non-symmetric convex domains have no basis of exponentials

1999/04/14 by Mihail N. Kolountzakis, Kolountzakis, Mihail N. · 1 citation
Mathematics · #42 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #math.CA #math.MG #msc:42

paper · pdf · doi:10.48550/arxiv.math/9904064

arxiv created 1999/12/18 · arxiv updated 2009/11/30

Abstract

A conjecture of Fuglede states that a bounded measurable set Ω in space, of measure 1, can tile space by translations if and only if the Hilbert space L²(Ω) has an orthonormal basis consisting of exponentials. If Ω has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain is not spectral.

Cited by

• Fuglede's conjecture holds for three intervals

Related

• A class of non-convex polytopes that admit no orthonormal basis of exponentials
• The Fuglede conjecture for convex domains is true in all dimensions
• Spectrality of product domains and Fuglede's conjecture for convex polytopes
• Convex bodies with a point of curvature do not have Fourier bases
• Fuglede conjecture holds for convex planar domains