The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

Karlheinz Gröchenig; Philippe Jaming

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arXiv · 1608.06738

The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

Karlheinz Gröchenig·Philippe Jaming

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Abstract

Two measurable sets $S, \Lambda \subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $\mu$ with support in S whose Fourier transform vanishes on {\Lambda} must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cram\'er-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients .

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BibTeX RIS

Karlheinz Gröchenig, Philippe Jaming. 2016-08-24. The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs. https://arxiv.org/abs/1608.06738

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