Heisenberg uniqueness pairs and the Klein-Gordon equation

• 成果类型: Article
• 署名作者: Hedenmalm, Haakan; Montes-Rodriguez, Alfonso
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.3.6
• 发表日期: 2011
• 页码: 1507-1527
• 关键词: even partial quotients; inner functions

摘要

A Heisenberg uniqueness pair (HUP) is a pair (Gamma, Lambda), where Gamma is a curve in the plane and Lambda is a set in the plane, with the following property: any finite Borel measure mu in the plane supported on Gamma, which is absolutely continuous with respect to arc length, and whose Fourier transform (mu) over cap vanishes on Lambda, must automatically be the zero measure. We prove that when is the hyperbola x(1) x(2) = 1 and Lambda is the lattice-cross Lambda = (alpha Z x {0}) boolean OR ({0} x beta Z) where alpha, beta are positive reals, then (Gamma,Lambda) is an HUP if and only if alpha,beta <= 1; in this situation, the Fourier transform (mu) over cap of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that e(pi i alpha nt) , e(pi i beta nt) , n is an element of Z w span a weak-star dense subspace in L-infinity( R) if and only if alpha beta <= 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.