Non-symmetric convex domains have no basis of exponentials
نویسنده
چکیده
A conjecture of Fuglede states that a bounded measurable set Ω ⊂ R, of measure 1, can tile R by translations if and only if the Hilbert space L(Ω) has an orthonormal basis consisting of exponentials eλ(x) = exp2πi〈λ, x〉. 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 in R is not spectral. §0. Introduction Let Ω be a measurable subset of Rd of measure 1 and Λ be a discrete subset of Rd. We write eλ(x) = exp 2πi〈λ, x〉, (x ∈ R ), EΛ = {eλ : λ ∈ Λ} ⊂ L (Ω). The inner product and norm on L(Ω) are 〈f, g〉 Ω = ∫
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