Riesz bases of exponentials and the Bohr topology

Cabrelli, Carlos; Hare, Kathryn; Molter, Ursula

doi:10.1090/proc/15395

Riesz bases of exponentials and the Bohr topology
HTML articles powered by AMS MathViewer

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