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Heisenberg uniqueness pairs and the Klein-Gordon equation
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). (Analysgruppen)ORCID-id: 0000-0002-4971-7147
Universidad de Sevilla. (Analiso Matematico)
2011 (engelsk)Inngår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 173, nr 3, s. 1507-1527Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

A Heisenberg uniqueness pair (HUP) is a pair (Γ,Λ), where Γ is a curve in the plane and Λ is a set in the plane, with the following property: any finite Borel measure μ in the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform μˆ vanishes on Λ, must automatically be the zero measure. We prove that when Γ is the hyperbola x1x2=1 %, and Λ is the lattice-cross Λ=(αZ×{0})∪({0}×βZ), where α,β are positive reals, then (Γ,Λ) is an HUP if and only if αβ≤1; in this situation, the Fourier transform μˆ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that eπiαnt,eπiβn/t,n∈Z, span a weak-star dense subspace in L∞(R) if and only if αβ≤1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

sted, utgiver, år, opplag, sider
2011. Vol. 173, nr 3, s. 1507-1527
Emneord [en]
composition operator, ergodic theory, inversion, Klein-Gordon equation, Trigonometric system
HSV kategori
Identifikatorer
URN: urn:nbn:se:kth:diva-60044DOI: 10.4007/annals.2011.173.3.6ISI: 000290722000006Scopus ID: 2-s2.0-79958820068OAI: oai:DiVA.org:kth-60044DiVA, id: diva2:477204
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QC 20120113

Tilgjengelig fra: 2012-01-12 Laget: 2012-01-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert

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