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2017

An optimal uncertainty principle in twelve dimensions via modular forms

Cohn, Henry, Gonçalves, Felipe

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We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane.

Suppose $f \colon \mathbb{R}^{12} \to \mathbb{R}$ is an integrable function that is not identically zero.
Normalize its Fourier transform $\widehat{f}$ by $\widehat{f}(\xi) = \int_{\mathbb{R}^d} f(x)e^{-2\pi i \langle x, \xi\rangle}\, dx$, and suppose $\widehat{f}$ is real-valued and integrable.
We show that if $f(0) \le 0$, $\widehat{f}(0) \le 0$, $f(x) \ge 0$ for $|x| \ge r_1$, and $\widehat{f}(\xi) \ge 0$ for $|\xi| \ge r_2$, then $r_1r_2 \ge 2$, and this bound is sharp.

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H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, Universal optimality of the E 8 E_{8} and Leech lattices and interpolation formulas , preprint, 2019. arXiv: 1902.05438
Original
1902
Earlier work this paper cites.
G. Szegő, Orthogonal Polynomials , fourth edition, AMS Colloquium Publications 23
1975
Earlier work this paper cites.
F. L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type (Russian), Algebra i Analiz 5
1994
Earlier work this paper cites.
T. C. Hales, Cannonballs and honeycombs , Notices Amer. Math. Soc. 47
2000
Earlier work this paper cites.
H. Cohn, New upper bounds on sphere packings II , Geom. Topol. 6
2002
Earlier work this paper cites.
H. Cohn and Y. Zhao, Sphere packing bounds via spherical codes , Duke Math. J. 163
Original
2002
Earlier work this paper cites.

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H. Cohn and S. D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 8 and 24 24 , preprint, 2016. arXiv: 1603.04759
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J. Bourgain, L. Clozel, and J.-P. Kahane, Principe d’Heisenberg et fonctions positives , Ann. Inst. Fourier (Grenoble) 60

2010

Cited alongside, same era.

D. Radchenko and M. Viazovska, Fourier interpolation on the real line , Publ. Math. Inst. Hautes Études Sci., to appear. doi: 10.1007/s10240-018-0101-z arXiv: 1701.00265

Cited in the paper.

2017

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H. Cohn and F. Gonçalves, Data for “An optimal uncertainty principle in twelve dimensions via modular forms” , data set, DSpace@MIT, 2018. http://hdl.handle.net/1721.1/118165

2018

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