Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Riesz bases of exponentials on multiband spectra


Author: Nir Lev
Journal: Proc. Amer. Math. Soc. 140 (2012), 3127-3132
MSC (2010): Primary 42C15, 94A12
DOI: https://doi.org/10.1090/S0002-9939-2012-11138-4
Published electronically: January 18, 2012
MathSciNet review: 2917085
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S$ be the union of finitely many disjoint intervals on $ \mathbb{R}$. Suppose that there are two real numbers $ \alpha , \beta $ such that the length of each interval belongs to $ \mathbb{Z} \alpha + \mathbb{Z}\beta $. We use quasicrystals to construct a discrete set $ \Lambda \subset \mathbb{R}$ such that the system of exponentials $ \{\exp 2 \pi i \lambda x, \, \lambda \in \Lambda \}$ is a Riesz basis in the space $ L^2(S)$.


References [Enhancements On Off] (What's this?)

  • 1. S. A. Avdonin, ``On the question of Riesz bases of exponential functions in $ L^2$'' (in Russian), Vestnik Leningrad Univ. 13 (1974), 5-12. English translation in Vestnik Leningrad Univ. Math. 7 (1979), 203-211. MR 0361746 (50:14191)
  • 2. L. Bezuglaya, V. Katsnelson, ``The sampling theorem for functions with limited multi-band spectrum'', Z. Anal. Anwendungen 12 (1993), 511-534. MR 1245936 (94k:94003)
  • 3. G. Kozma, N. Lev, ``Exponential Riesz bases, discrepancy of irrational rotations and BMO'', J. Fourier Anal. Appl. 17 (2011), 879-898.
  • 4. Yu. Lyubarskii, K. Seip, ``Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets'', J. Fourier Anal. Appl. 3 (1997), 597-615. MR 1491937 (99f:30007)
  • 5. Yu. Lyubarskii, I. Spitkovsky, ``Sampling and interpolation for a lacunary spectrum'', Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 77-87. MR 1378833 (97b:41004)
  • 6. B. Matei, Y. Meyer, ``Quasicrystals are sets of stable sampling'', C. R. Acad. Sci. Paris, Ser. I 346 (2008), 1235-1238. MR 2473299 (2010g:94050)
  • 7. B. Matei, Y. Meyer, ``Simple quasicrystals are sets of stable sampling'', Complex Var. Elliptic Equ. 55 (2010), 947-964. MR 2674875
  • 8. K. Seip, ``A simple construction of exponential bases in $ L^2$ of the union of several intervals'', Proc. Edinburgh Math. Soc. 38 (1995), 171-177. MR 1317335 (96c:42006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42C15, 94A12

Retrieve articles in all journals with MSC (2010): 42C15, 94A12


Additional Information

Nir Lev
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: nir.lev@weizmann.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2012-11138-4
Keywords: Riesz bases, multiband signals, quasicrystals
Received by editor(s): February 6, 2011
Received by editor(s) in revised form: March 21, 2011
Published electronically: January 18, 2012
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society