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  • Karlheinz Gröchenig (a1) and Philippe Jaming (a2) (a3)

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$ . As a corollary we obtain a new, surprising version of the classical Cramér–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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1. Babot, D. B., Heisenberg uniqueness pairs in the plane. Three parallel lines, Proc. Amer. Math. Soc. 141 (2013), 38993904.
2. Bélisle, C., Massé, J.-C. and Ransford, Th., When is a probability measure determined by infinitely many projections, Ann. Probab. 25 (1997), 767786.
3. Cramér, H. and Wold, H., Some theorems on distribution functions, J. Lond. Math. Soc. (2) 11 (1936), 290294.
4. Dellacherie, C. and Meyer, P. A., Probabilities and Potential (North Holland, Amsterdam, 1978).
5. Fernández-Bertolin, A., Gröchenig, K. and Jaming, Ph., Heisenberg uniqueness pairs and unique continuation for harmonic functions and solutions of the Helmholtz equation. In preparation, technical report.
6. Gilbert, W. M., Projections of probability distributions, Acta Math. Acad. Sci. Hungar. 6 (1955), 195198.
7. Hedenmalm, H. and Montes-Rodríguez, A., Heisenberg uniqueness pairs and the Klein–Gordon equation, Ann. of Math. (2) 173 (2011), 15071527.
8. Heppes, A., On the determination of probability distributions of more dimensions by their projections, Acta Math. Acad. Sci. Hungar. 7 (1956), 403410.
9. Hohlweg, C., Labbé, J.-Ph. and Ripoll, V., Asymptotical behaviour of roots of infinite Coxeter groups, Canad. J. Math. 66 (2014), 323353.
10. Jaming, Ph. and Kellay, K., A dynamical system approach to Heisenberg uniqueness pairs. J. Anal. Math., to appear, available on arXiv:arXiv:1312.6236.
11. Lev, N., Uniqueness theorems for Fourier transforms, Bull. Sci. Math. 135 (2011), 134140.
12. Radin, C. and Sadum, L., On 2-generator subgroups of SO(3), Trans. Amer. Math. Soc. 351 (1999), 44694480.
13. Rényi, A., On projections of probability distributions, Acta Math. Acad. Sci. Hungar. 3 (1952), 131142.
14. Sjölin, P., Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin, Bull. Sci. Math. 135 (2011), 125133.
15. Sjölin, P., Heisenberg uniqueness pairs for the parabola, J. Fourier Anal. Appl. 19 (2013), 410416.
16. Giri, D. K. and Srivastava, R. K., Heisenberg uniqueness pairs for some algebraic curves in the plane, Adv. Math. 310 (2017), 9931016.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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