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Heisenberg uniqueness pairs and the Klein-Gordon equation
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Analysgruppen)ORCID iD: 0000-0002-4971-7147
Universidad de Sevilla. (Analiso Matematico)
2011 (English)In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 173, no 3, 1507-1527 p.Article in journal (Refereed) 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.

Place, publisher, year, edition, pages
2011. Vol. 173, no 3, 1507-1527 p.
Keyword [en]
composition operator, ergodic theory, inversion, Klein-Gordon equation, Trigonometric system
National Category
URN: urn:nbn:se:kth:diva-60044DOI: 10.4007/annals.2011.173.3.6ISI: 000290722000006ScopusID: 2-s2.0-79958820068OAI: diva2:477204

QC 20120113

Available from: 2012-01-12 Created: 2012-01-12 Last updated: 2013-01-22Bibliographically approved

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Hedenmalm, Håkan
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