Riesz bases of exponentials and the Bohr topology
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- by Carlos Cabrelli, Kathryn E. Hare and Ursula Molter PDF
- Proc. Amer. Math. Soc. 149 (2021), 2121-2131 Request permission
Abstract:
We provide a necessary and sufficient condition to ensure that a multi-tile $\Omega \subset \mathbb {R}^{d}$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $L^{2}(\Omega )$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.References
- 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, DOI 10.1016/j.aim.2015.08.006
- Carlos Cabrelli and Diana Carbajal, Riesz bases of exponentials on unbounded multi-tiles, Proc. Amer. Math. Soc. 146 (2018), no. 5, 1991–2004. MR 3767351, DOI 10.1090/proc/13980
- Anton Deitmar and Siegfried Echterhoff, Principles of harmonic analysis, Universitext, Springer, New York, 2009. MR 2457798
- J. Feldman and F. P. Greenleaf, Existence of Borel transversals in groups, Pacific J. Math. 25 (1968), 455–461. MR 230837, DOI 10.2140/pjm.1968.25.455
- Colin C. Graham and Kathryn E. Hare, Interpolation and Sidon sets for compact groups, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2013. MR 3025283, DOI 10.1007/978-1-4614-5392-5
- Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI 10.1016/0022-1236(74)90072-x
- Sigrid Grepstad and Nir Lev, Multi-tiling and Riesz bases, Adv. Math. 252 (2014), 1–6. MR 3144222, DOI 10.1016/j.aim.2013.10.019
- Sigrid Grepstad and Nir Lev, Riesz bases, Meyer’s quasicrystals, and bounded remainder sets, Trans. Amer. Math. Soc. 370 (2018), no. 6, 4273–4298. MR 3811528, DOI 10.1090/tran/7157
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- A. Iosevich, Fuglede conjecture for lattices, preprint available at www.math.rochester.edu/people/faculty/iosevich/expository/FugledeLattice.pdf.
- Gady Kozma and Shahaf Nitzan, Combining Riesz bases, Invent. Math. 199 (2015), no. 1, 267–285. MR 3294962, DOI 10.1007/s00222-014-0522-3
- Gady Kozma and Shahaf Nitzan, Combining Riesz bases in $\Bbb {R}^d$, Rev. Mat. Iberoam. 32 (2016), no. 4, 1393–1406. MR 3593529, DOI 10.4171/RMI/922
- Mihail N. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials, Proc. Amer. Math. Soc. 143 (2015), no. 2, 741–747. MR 3283660, DOI 10.1090/S0002-9939-2014-12310-0
- Mihail N. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), no. 3, 542–550. MR 1772427
- Mihail N. Kolountzakis, The study of translational tiling with Fourier Analysis, Lectures given at the Workshop on Fourier Analysis and Convexity, Universita di Milano-Bicocca, pp. 11-22, (2001).
- Eberhard Kaniuth and Gitta Kutyniok, Zeros of the Zak transform on locally compact abelian groups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3561–3569. MR 1459128, DOI 10.1090/S0002-9939-98-04450-5
- I. Łaba, Fuglede’s conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2965–2972. MR 1840101, DOI 10.1090/S0002-9939-01-06035-X
- J. Marzo, Riesz basis of exponentials for a union of cubes in $\mathbb {R}^{d}$, arXiv:math/0601288, (2006).
- Alexander M. Olevskii and Alexander Ulanovskii, Functions with disconnected spectrum, University Lecture Series, vol. 65, American Mathematical Society, Providence, RI, 2016. Sampling, interpolation, translates. MR 3468930, DOI 10.1090/ulect/065
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- Robert M. Young, An introduction to nonharmonic Fourier series, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR 1836633
Additional Information
- Carlos Cabrelli
- Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina
- MR Author ID: 308374
- ORCID: 0000-0002-6473-2636
- Email: carlos.cabrelli@gmail.com
- Kathryn E. Hare
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada
- MR Author ID: 246969
- Email: kehare@uwaterloo.ca
- Ursula Molter
- Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina
- MR Author ID: 126270
- Email: umolter@dm.uba.ar
- Received by editor(s): June 6, 2020
- Received by editor(s) in revised form: September 24, 2020
- Published electronically: February 23, 2021
- Additional Notes: The research of the first and third authors was partially supported by Grants: CONICET PIP 11220110101018, PICT-2014-1480, UBACyT 20020130100403BA, UBACyT 20020130100422B
The research of the second author was partially supported by NSERC 2016-03719. - Communicated by: Dmitriy Bilyk
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 2121-2131
- MSC (2020): Primary 42B99, 42C15; Secondary 42A10, 05B45, 42A15
- DOI: https://doi.org/10.1090/proc/15395
- MathSciNet review: 4232203