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Heisenberg uniqueness pairs in the plane. Three parallel lines

Author: Daniel Blasi Babot
Journal: Proc. Amer. Math. Soc. 141 (2013), 3899-3904
MSC (2010): Primary 42B10; Secondary 31B35, 81Q05
Published electronically: July 18, 2013
MathSciNet review: 3091778
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Abstract: A Heisenberg uniqueness pair 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 bounded Borel measure $ \mu $ in the plane supported on $ \Gamma ,$ which is absolutely continuous with respect to arc length and whose Fourier transform $ \widehat {\mu }$ vanishes on $ \Lambda ,$ must automatically be the zero measure. We characterize the Heisenberg uniqueness pairs for $ \Gamma $ as being three parallel lines $ \Gamma =\mathbb{R}\times \{\alpha ,\beta ,\gamma \}$ with $ \alpha <\beta <\gamma ,$ $ (\gamma -\alpha )/(\beta -\alpha )\in \mathbb{N}.$

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Additional Information

Daniel Blasi Babot
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Catalunya, Spain

Keywords: Heisenberg uniqueness pairs, uncertainty principle
Received by editor(s): March 29, 2011
Received by editor(s) in revised form: October 29, 2011, and January 23, 2012
Published electronically: July 18, 2013
Additional Notes: The author was partially supported by the Göran Gustaffson Foundation, grant No. 2009SGR00420, and the DGICYT grant MTM2008-00145
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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