Unions of exponential Riesz bases

Dae Gwan Lee; Dae Gwan Lee

doi:10.3934/math.20241161

AIMS Mathematics

2024, Volume 9, Issue 9: 23890-23908. doi: 10.3934/math.20241161

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Research article

Unions of exponential Riesz bases

Dae Gwan Lee ^,

Department of Mathematics and Big Data Science, Kumoh National Institute of Technology, Gumi, Gyeongsangbuk-do 39177, Korea

Received: 09 February 2024 Revised: 06 April 2024 Accepted: 15 April 2024 Published: 12 August 2024
MSC : 42C15

We have developed new methods for constructing exponential Riesz bases by combining existing ones. These methods involve taking unions of frequency sets and domains respectively, offering easier construction compared to known techniques. Along with examples illustrating our methods, we also provide several examples that highlight the intricate nature of exponential Riesz bases.
- complex exponentials,
- Riesz bases,
- frames,
- spectrum,
- domain,
- union
Citation: Dae Gwan Lee. Unions of exponential Riesz bases[J]. AIMS Mathematics, 2024, 9(9): 23890-23908. doi: 10.3934/math.20241161

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Abstract

We have developed new methods for constructing exponential Riesz bases by combining existing ones. These methods involve taking unions of frequency sets and domains respectively, offering easier construction compared to known techniques. Along with examples illustrating our methods, we also provide several examples that highlight the intricate nature of exponential Riesz bases.

References

[1]	M. Bownik, P. Casazza, A. W. Marcus, D. Speegle, Improved bounds in Weaver and Feichtinger conjectures, J. Reine Angew. Math., 749 (2019), 267–293. https://doi.org/10.1515/crelle-2016-0032 doi: 10.1515/crelle-2016-0032
[2]	C. Cabrelli, D. Carbajal, Riesz bases of exponentials on unbounded multi-tiles, Proc. Amer. Math. Soc., 146 (2018), 1991–2004. https://doi.org/10.1090/proc/13980 doi: 10.1090/proc/13980
[3]	A. Caragea, D. G. Lee, A note on exponential Riesz bases, Sampl. Theory Signal Process. Data Anal., 20 (2022), 13. https://doi.org/10.1007/s43670-022-00031-9 doi: 10.1007/s43670-022-00031-9
[4]	O. Christensen, An Introduction to Frames and Riesz Bases, 2 Eds., Boston: Birkhäuser, 2016. https://doi.org/10.1007/978-3-319-25613-9
[5]	L. De Carli, Concerning exponential bases on multi-rectangles of $ \mathbb R^d$, In: Topics in Classical and Modern Analysi, Applied and Numerical Harmonic Analysis, Boston: Birkhäuser, 2019, 65–85. https://doi.org/10.1007/978-3-030-12277-5
[6]	A. Debernardi, N. Lev, Riesz bases of exponentials for convex polytopes with symmetric faces, J. Eur. Math. Soc., 24 (2022), 3017–3029. https://doi.org/10.4171/jems/1158 doi: 10.4171/jems/1158
[7]	C. Frederick, A. Mayeli, Frame spectral pairs and exponential bases, J. Four. Anal. Appl., 27 (2021), 75. https://doi.org/10.1007/s00041-021-09872-9 doi: 10.1007/s00041-021-09872-9
[8]	S. Grepstad, N. Lev, Multi-tiling and Riesz bases, Adv. Math., 252 (2014), 1–6. https://doi.org/10.1016/j.aim.2013.10.019
[9]	C. Heil, History and evolution of the density theorem for Gabor frames, J. Four. Anal. Appl., 12 (2007), 113–166. https://doi.org/10.1007/s00041-006-6073-2 doi: 10.1007/s00041-006-6073-2
[10]	R. A. Horn, C. R. Johnson, Matrix Analysis, 2 Eds., New York: Cambridge University Press, 2013.
[11]	M. N. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials, Proc. Amer. Math. Soc., 143 (2015), 741–747. https://doi.org/10.1090/S0002-9939-2014-12310-0 doi: 10.1090/S0002-9939-2014-12310-0
[12]	G. Kozma, S. Nitzan, Combining Riesz bases, Invent. Math., 199 (2015), 267–285. https://doi.org/10.1007/s00222-014-0522-3 doi: 10.1007/s00222-014-0522-3
[13]	G. Kozma, S. Nitzan, A. Olevskii, A set with no Riesz basis of exponentials, Rev. Mat. Iberoam., 39 (2023), 2007–2016. https://doi.org/10.4171/rmi/1411 doi: 10.4171/rmi/1411
[14]	H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117 (1967), 37–52. https://doi.org/10.1007/BF02395039 doi: 10.1007/BF02395039
[15]	D. G. Lee, On construction of bounded sets not admitting a general type of Riesz spectrum, Axioms, 13 (2024), 36. https://doi.org/10.3390/axioms13010036 doi: 10.3390/axioms13010036
[16]	D. G. Lee, G. E. Pfander, D. Walnut, Bases of complex exponentials with restricted supports, J. Math. Anal. Appl., 521 (2023), 126917. https://doi.org/10.1016/j.jmaa.2022.126917 doi: 10.1016/j.jmaa.2022.126917
[17]	D. G. Lee, G. E. Pfander, D. Walnut, Cube tilings with linear constraints, Results Math., 79 (2024), 212. https://doi.org/10.1007/s00025-024-02243-y doi: 10.1007/s00025-024-02243-y
[18]	D. G. Lee, G. E. Pfander, D. Walnut, Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice, preprint paper, 2024. https://doi.org/10.48550/arXiv.2401.08042
[19]	N. Lev, Riesz bases of exponentials on multiband spectra, Proc. Amer. Math. Soc., 140 (2012), 3127–3132. https://doi.org/10.1090/S0002-9939-2012-11138-4 doi: 10.1090/S0002-9939-2012-11138-4
[20]	Y. Lyubarskii, A. Rashkovskii, Complete interpolation sequences for Fourier transforms supported by convex symmetric polygons, Ark. Mat., 38 (2000), 139–170. https://doi.org/10.1007/BF02384495 doi: 10.1007/BF02384495
[21]	Y. Lyubarskii, K. Seip, A splitting problem for unconditional bases of complex exponentials, In: Entire functions in modern analysis: Boris Levin Memorial Conference (Tel-Aviv, 1997) (eds. Y. Lyubich et al.), Israel Math. Conf. Proc. vol. 15, Bar-Ilan Univ., Ramat Gan, 2001,233–242.
[22]	J. Marzo, Riesz basis of exponentials for a union of cubes in $ \mathbb R^d$, preprint paper, 2006. https://doi.org/10.48550/arXiv.math/0601288
[23]	B. Matei, Y. Meyer, A variant of compressed sensing, Rev. Mat. Iberoam., 25 (2009), 669–692. https://doi.org/10.4171/rmi/578 doi: 10.4171/rmi/578
[24]	A. Olevskii, A. Ulanovskii, Functions with Disconnected Spectrum: Sampling, Interpolation, Translates, New York: American Mathematical Society, University Lecture Series, 2016.
[25]	G. E. Pfander, S. Revay, D. Walnut, Exponential bases for partitions of intervals, Appl. Comput. Harmon. Anal., 68 (2024), 101607. https://doi.org/10.1016/j.acha.2023.101607 doi: 10.1016/j.acha.2023.101607
[26]	K. Seip, A simple construction of exponential bases in $L^2$ of the union of several intervals, Proc. Edinb. Math. Soc., 38 (1995), 171–177. https://doi.org/10.1017/S0013091500006295 doi: 10.1017/S0013091500006295
[27]	P. Stevenhagen, H. W. Lenstra, Chebotarëv and his density theorem, Math. Intell., 18 (1996), 26–37. https://doi.org/10.1007/BF03027290 doi: 10.1007/BF03027290
[28]	R. M. Young, An Introduction to Nonharmonic Fourier Series, New York: Academic Press, 2001.

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