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Frame Spectral Pairs and Exponential Bases

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Abstract

Given a domain \(\varOmega \subset {\mathbb {R}}^d\) with positive and finite Lebesgue measure and a discrete set \(\varLambda \subset {\mathbb {R}}^d\), we say that \((\varOmega , \varLambda )\) is a frame spectral pair if the set of exponential functions \({\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}\) is a frame for \(L^2(\varOmega )\). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting \({\mathbb {Z}}_N^d\), \(d, N\ge 1\), a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in \({\mathbb {R}}^d\) by “adding” a frame spectral pair in \({\mathbb {R}}^{d}\) to a frame spectral pair in \({\mathbb {Z}}_N^d\). Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.

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Notes

  1. Note that the definition of this general function over \({\mathbb {R}}\), also called the symbol of J, coincides with the Fourier transform of the characteristic function \(\chi _J\) over the cyclic group when the domain is restricted to \({\mathbb {Z}}_N\) up to a constant.

  2. By (30d), the Fourier transform of \(F_s\) is equal to a sum of translated copies of the Fourier transform of f on k-shifts of \(\varOmega \) multiplied with coefficients \({\hat{\chi }}_J(k)\). The theorem proves that the Fourier transform of \(F_s\) over \(\varOmega \) is the exact Fourier transform of f up to some constant, and the translations do not overlap. In the language of signal processing, this means that the aliasing term is zero.

References

  1. Agora, E., Antezana, J., Cabrelli, C.: Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups. Adv. Math. 285, 454–477 (2015)

    Article  MathSciNet  Google Scholar 

  2. Birklbauer, P.: Fuglede Conjecture holds in \({\mathbb{Z}}_5^3\)> Experimental Mathematics, published online: 12 Jul 2019. https://doi.org/10.1080/10586458.2019.1636427

  3. Cabrelli, C., Carbajal, D.: Riesz bases of exponentials on unbounded multi-tiles. Proc. Am. Math. Soc. 146, 1991–2004 (2018)

    Article  MathSciNet  Google Scholar 

  4. Casazza, P.G.: The art of frame theory. Taiwan. J. Math. 4, 129–201 (2000)

    Article  MathSciNet  Google Scholar 

  5. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2016)

    MATH  Google Scholar 

  6. De Carli, L.: Exponential bases on multi-rectangles in \({\mathbb{R}}^{d}\), preprint, arXiv:1512.02275 (2015)

  7. Debernardi, A., Lev, N.: Riesz bases of exponentials for convex polytopes with symmetric faces, preprint (2019)

  8. Debernardi, A., Lev, N.: Riesz bases of exponentials for convex polytopes with symmetric faces, preprint, arXiv:1907.04561

  9. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  10. Fallon, T., Kiss, G., Somlai, G.: Spectral sets and tiles in \({\mathbb{Z}} _p^2\times {\mathbb{Z}} _q^2\), preprint, arXiv:2105.10575

  11. Fallon,T., Mayeli, A., Villano, D.: The Fuglede Conjecture holds in \({\mathbb{F}} _p^3\) for p=5,7, to appear in Proceeding of AMS, https://doi.org/10.1090/proc/14750

  12. Ferguson, S., Mayeli, A., Sothanaphan, N.: Exponential Riesz bases, multi-tiling and condition numbers in finite abelian groups, preprint arXiv:1904.04487

  13. Frederick, C., Okoudjou, K.: Finding duality and Riesz bases of exponentials on multi-tiles. Appl. Comput. Harmon. Anal. 51, 104–117 (2021)

    Article  MathSciNet  Google Scholar 

  14. Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)

    Article  MathSciNet  Google Scholar 

  15. Fuglede, B.: Orthogonal exponentials on the ball. Expo. Math. 19, 267–272 (2001)

    Article  MathSciNet  Google Scholar 

  16. Greenfeld, R., Lev, N.: Spectrality of product domains and Fuglede’s conjecture for convex polytopes. J. Anal. Math. 140(2), 409–441 (2020)

    Article  MathSciNet  Google Scholar 

  17. Grepstad, R., Lev, N.: Multi-tiling and Riesz bases. Adv. Math. 252(15), 1–6 (2014)

    Article  MathSciNet  Google Scholar 

  18. Iosevich, A., Kolountzakis, M.: Size of orthogonal sets of exponentials for the disk. Rev. Mat. Iberoam. 29(2), 739–747 (2013)

    Article  MathSciNet  Google Scholar 

  19. Iosevich, A., Katz, N.H., Tao, T.: Convex bodies with a point of curvature do not have Fourier bases. Am. J. Math. 123, 115–120 (2001)

    Article  MathSciNet  Google Scholar 

  20. Iosevich, A., Katz, N., Tao, T.: The Fuglede spectral conjecture holds for convex planar domains. Math. Res. Lett. 10(5–6), 559–569 (2003)

    Article  MathSciNet  Google Scholar 

  21. Iosevich, A., Mayeli, A., Pakianathan, J.: The Fuglede Conjecture holds in \(\mathbb{Z}_p\times \mathbb{Z}_p\). Anal. PDE 10(4), 757–764 (2017)

    Article  MathSciNet  Google Scholar 

  22. Jorgensen, P.E.T., Pedersen, S.: Spectral theory for Borel sets in Rn of finite measure. J. Funct. Anal. 107, 72–104 (1992)

    Article  MathSciNet  Google Scholar 

  23. Jorgensen, P.E.T., Pedersen, S.: Harmonic analysis and fractal limit-measures induced by representations of a certain \(C^\ast \)-algebra. J. Funct. Anal. 125, 90–110 (1994)

    Article  MathSciNet  Google Scholar 

  24. Jorgensen, P., Pedersen, S.: Spectral pairs in Cartesian coordinates. J. Fourier Anal. Appl. 5(4), 285–302 (1999)

    Article  MathSciNet  Google Scholar 

  25. Kolountzakis, M.: Non-symmetric convex domains have no basis of exponentials. Illinois J. Math. 44(3) (1999)

  26. Kolountzakis, M.N.: Multiple lattice tiles and Riesz bases of exponentials. Proc. Am. Math. Soc. 143(2), 741–747 (2015)

    Article  MathSciNet  Google Scholar 

  27. Kolountzakis, M., Matolcsi, M.: Complex Hadamard matrices and the spectral set conjecture, Collectanea Mathematica, Supl. pp. 282–291 (2006)

  28. Kolountzakis, M., Matolcsi, M.: Tiles with no spectra. Forum Math. 18(3), 519–528 (2006)

    Article  MathSciNet  Google Scholar 

  29. Kozma, G., Nitzan, S.: Combining Riesz bases in \(\mathbb{R}^d\). Rev. Mate. Iberoam. 32(4), 1393–1406 (2016)

    Article  Google Scholar 

  30. Łaba, I.: Fuglede’s conjecture for a union of two intervals. Proc. Am. Math. Soc. 129(10), 2965–2972 (2001)

    Article  MathSciNet  Google Scholar 

  31. Lyubarskii, Y., Seip, K.: Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets. J. Fourier Anal. Appl. 3, 597–615 (1997)

    Article  MathSciNet  Google Scholar 

  32. Marzo, J.: Riesz basis of exponentials for a union of cubes in \({\mathbb{R}}^{d}\). arXiv preprint math/0601288. 2006 Jan 12

  33. Matolcsi, M.: Fuglede conjecture fails in dimension \(4\). Proc. Am. Math. Soc. 133(10), 3021–3026 (2005)

    Article  MathSciNet  Google Scholar 

  34. Nitzan, S., Olevskii, A., Ulanovskii, A.: Exponential frames on unbounded sets. Proc. Am. Math. Soc. 144, 109–118 (2016)

    Article  MathSciNet  Google Scholar 

  35. Tao, T.: Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11(2–3), 251–258 (2004)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, USA

    Christina Frederick

  2. Department of Mathematics, City University of New York, New York, USA

    Azita Mayeli

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  1. Christina Frederick
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  2. Azita Mayeli
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Correspondence to Christina Frederick.

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Communicated by Chris Heil.

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Frederick, C., Mayeli, A. Frame Spectral Pairs and Exponential Bases. J Fourier Anal Appl 27, 75 (2021). https://doi.org/10.1007/s00041-021-09872-9

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  • DOI: https://doi.org/10.1007/s00041-021-09872-9

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