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Perron-Frobenius operators and the Klein-Gordon equation

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Autor: Canto Martín, Francisco Manuel
Hedenmalm, Håkan
Montes Rodríguez, Alfonso
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2014
Publicado en: Journal of the European Mathematical Society, 16 (1), 31-66.
Tipo de documento: Artículo
Resumen: For a smooth curve Γ and a set Λ in the plane R2, let AC(Γ; Λ) be the space of finite Borel measures in the plane supported on Γ, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ. Following [12], we say that (Γ, Λ) is a Heisenberg uniqueness pair if AC(Γ; Λ) = {0}. In the context of a hyperbola Γ, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Γ; Λ) when it is nonzero. We will fix the curve Γ to be the hyperbola x1x2 = 1, and the set Λ = Λα,β to be the lattice-cross Λα,β = (αZ × {0}) ∪ ({0} × βZ), where α, β are positive reals. We will also consider Γ+, the branch of x1x2 = 1 where x1 > 0. In [12], it is shown that AC(Γ; Λα,β) = {0} if and only if αβ ≤ 1. Here, we show that for αβ > 1, we get a rather drastic “phase transition”: AC(Γ; Λα,β) is infinite-dimensional whene...
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Cita: Canto Martín, F.M., Hedenmalm, H. y Montes Rodríguez, A. (2014). Perron-Frobenius operators and the Klein-Gordon equation. Journal of the European Mathematical Society, 16 (1), 31-66.
Tamaño: 385.2Kb
Formato: PDF

URI: http://hdl.handle.net/11441/45142

DOI: 10.4171/JEMS/427

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