Set Spectrum

A spectrum of a measurable set with finite positive Lebesgue measure is a set for which

${e^(2piix·lambda)}_(lambda in Lambda)$

is an orthogonal basis of the Hilbert space . A set that admits a spectrum is called a spectral set. Fuglede's conjecture concerns the relationship between spectral sets and sets that tile by translations.

See also

Fuglede's Conjecture, Orthogonal Basis

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References

Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974. https://doi.org/10.1016/0022-1236(74)90072-X.Iosevich, A.; Katz, N. H.; and Tao, T. "Convex Bodies with a Point of Curvature Do Not Have Fourier Bases." Amer. J. Math. 123, 115-120, 2001. https://doi.org/10.1353/ajm.2001.0003.

Cite this as:

Weisstein, Eric W. "Set Spectrum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SetSpectrum.html

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