spektraalset

A spektraalset, often called a spectral set, is a measurable subset E of Euclidean space R^n with finite positive measure that admits a spectrum. Concretely, E is spectral if there exists a countable set Λ ⊂ R^n such that the family of complex exponentials {e^{2π i ⟨λ, x⟩} : λ ∈ Λ} forms an orthonormal basis for the Hilbert space L^2(E). In this situation, E is called spectral and Λ is called a spectrum for E; the pair (E, Λ) is a spectral pair.

A key example is the unit cube E = [0,1]^n. The spectrum Λ = Z^n yields the usual Fourier basis {e^{2π i ⟨k, x⟩} : k ∈ Z^n}, which forms an orthonormal basis for L^2([0,1]^n). Other spectral sets include certain finite unions of intervals in one dimension, and various constructions obtained by lattice tilings and product structures.

Fuglede's conjecture proposed that a measurable set is spectral if and only if it tiles R^n by translations. The conjecture sparked extensive study and many partial results, but it is known to be false in general for dimensions five and higher: there exist spectral sets that do not admit a translation tiling, and conversely, tiling sets that are not spectral have also been explored. In lower dimensions, the problem is more nuanced and remains unsettled in full generality, although complete classifications exist for some simple cases (such as intervals and certain unions of intervals) and for the one-dimensional setting in specific families.

Overall, the concept of spektraalset sits at the intersection of harmonic analysis, geometry, and tiling theory, linking spectral properties of function spaces to translational tiling phenomena.