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Riesz bases of exponentials on unbounded multi-tiles


Authors: Carlos Cabrelli and Diana Carbajal
Journal: Proc. Amer. Math. Soc. 146 (2018), 1991-2004
MSC (2010): Primary 42B99, 42C15; Secondary 42A10, 42A15
DOI: https://doi.org/10.1090/proc/13980
Published electronically: January 29, 2018
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Abstract: We prove the existence of Riesz bases of exponentials of $ L^2(\Omega )$, provided that $ \Omega \subset \mathbb{R}^d$ is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case.


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  • [1] Elona Agora, Jorge Antezana, and Carlos Cabrelli, Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups, Adv. Math. 285 (2015), 454-477. MR 3406506
  • [2] Boris Alexeev, Jameson Cahill, and Dustin G. Mixon, Full spark frames, J. Fourier Anal. Appl. 18 (2012), no. 6, 1167-1194. MR 3000979
  • [3] Marcin Bownik, The structure of shift-invariant subspaces of $ L^2({\bf R}^n)$, J. Funct. Anal. 177 (2000), no. 2, 282-309. MR 1795633
  • [4] D. Barbieri, C. Cabrelli, E. Hernández, P. Luthy, U. Molter and C. Mosquera, C. R. Math. Acad. Sci. Paris, to appear 2018.
  • [5] Davide Barbieri, Eugenio Hernández, and Azita Mayeli, Lattice sub-tilings and frames in LCA groups, C. R. Math. Acad. Sci. Paris 355 (2017), no. 2, 193-199. MR 3612708
  • [6] Carlos Cabrelli and Victoria Paternostro, Shift-invariant spaces on LCA groups, J. Funct. Anal. 258 (2010), no. 6, 2034-2059. MR 2578463
  • [7] Bálint Farkas, Máté Matolcsi, and Péter Móra, On Fuglede's conjecture and the existence of universal spectra, J. Fourier Anal. Appl. 12 (2006), no. 5, 483-494. MR 2267631
  • [8] Bálint Farkas and Szilárd Gy. Révész, Tiles with no spectra in dimension 4, Math. Scand. 98 (2006), no. 1, 44-52. MR 2221543
  • [9] Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101-121. MR 0470754
  • [10] Sigrid Grepstad and Nir Lev, Multi-tiling and Riesz bases, Adv. Math. 252 (2014), 1-6. MR 3144222
  • [11] Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
  • [12] A. Iosevich, Fuglede conjecture for lattices, preprint available at www.math.rochester.edu
    /people/faculty/iosevich/expository/FugledeLattice.pdf.
  • [13] Alex Iosevich, Nets Katz, and Terence Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), no. 5-6, 559-569. MR 2024715
  • [14] Gady Kozma and Shahaf Nitzan, Combining Riesz bases, Invent. Math. 199 (2015), no. 1, 267-285. MR 3294962
  • [15] Gady Kozma and Shahaf Nitzan, Combining Riesz bases in $ \mathbb{R}^d$, Rev. Mat. Iberoam. 32 (2016), no. 4, 1393-1406. MR 3593529
  • [16] Mihail N. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials, Proc. Amer. Math. Soc. 143 (2015), no. 2, 741-747. MR 3283660
  • [17] Mihail N. Kolountzakis and Máté Matolcsi, Tiles with no spectra, Forum Math. 18 (2006), no. 3, 519-528. MR 2237932
  • [18] Mihail N. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), no. 3, 542-550. MR 1772427
  • [19] Basarab Matei and Yves Meyer, Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 947-964. MR 2674875
  • [20] Basarab Matei and Yves Meyer, Quasicrystals are sets of stable sampling, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1235-1238 (English, with English and French summaries). MR 2473299
  • [21] Shahaf Nitzan, Alexander Olevskii, and Alexander Ulanovskii, Exponential frames on unbounded sets, Proc. Amer. Math. Soc. 144 (2016), no. 1, 109-118. MR 3415581
  • [22] Kristian Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, vol. 33, American Mathematical Society, Providence, RI, 2004. MR 2040080
  • [23] Terence Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251-258. MR 2067470
  • [24] Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121-127. MR 2122735
  • [25] Robert M. Young, An introduction to nonharmonic Fourier series, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR 1836633

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Additional Information

Carlos Cabrelli
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina
Email: cabrelli@dm.uba.ar

Diana Carbajal
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina
Email: dcarbajal@dm.uba.ar

DOI: https://doi.org/10.1090/proc/13980
Keywords: Riesz bases of exponentials, frames of exponentials, multi-tiling, submulti-tiling, Paley-Wiener spaces, shift-invariant spaces
Received by editor(s): January 24, 2017
Received by editor(s) in revised form: May 8, 2017
Published electronically: January 29, 2018
Additional Notes: The research of the authors was partially supported by Grants: CONICET PIP 11220110101018, PICT-2014-1480, UBACyT 20020130100403BA, UBACyT 20020130100422BA.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2018 American Mathematical Society

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