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
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.
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
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)
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
Cabrelli, C., Carbajal, D.: Riesz bases of exponentials on unbounded multi-tiles. Proc. Am. Math. Soc. 146, 1991–2004 (2018)
Casazza, P.G.: The art of frame theory. Taiwan. J. Math. 4, 129–201 (2000)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2016)
De Carli, L.: Exponential bases on multi-rectangles in \({\mathbb{R}}^{d}\), preprint, ar**v:1512.02275 (2015)
Debernardi, A., Lev, N.: Riesz bases of exponentials for convex polytopes with symmetric faces, preprint (2019)
Debernardi, A., Lev, N.: Riesz bases of exponentials for convex polytopes with symmetric faces, preprint, ar**v:1907.04561
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Fallon, T., Kiss, G., Somlai, G.: Spectral sets and tiles in \({\mathbb{Z}} _p^2\times {\mathbb{Z}} _q^2\), preprint, ar**v:2105.10575
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
Ferguson, S., Mayeli, A., Sothanaphan, N.: Exponential Riesz bases, multi-tiling and condition numbers in finite abelian groups, preprint ar**v:1904.04487
Frederick, C., Okoudjou, K.: Finding duality and Riesz bases of exponentials on multi-tiles. Appl. Comput. Harmon. Anal. 51, 104–117 (2021)
Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)
Fuglede, B.: Orthogonal exponentials on the ball. Expo. Math. 19, 267–272 (2001)
Greenfeld, R., Lev, N.: Spectrality of product domains and Fuglede’s conjecture for convex polytopes. J. Anal. Math. 140(2), 409–441 (2020)
Grepstad, R., Lev, N.: Multi-tiling and Riesz bases. Adv. Math. 252(15), 1–6 (2014)
Iosevich, A., Kolountzakis, M.: Size of orthogonal sets of exponentials for the disk. Rev. Mat. Iberoam. 29(2), 739–747 (2013)
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)
Iosevich, A., Katz, N., Tao, T.: The Fuglede spectral conjecture holds for convex planar domains. Math. Res. Lett. 10(5–6), 559–569 (2003)
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)
Jorgensen, P.E.T., Pedersen, S.: Spectral theory for Borel sets in Rn of finite measure. J. Funct. Anal. 107, 72–104 (1992)
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)
Jorgensen, P., Pedersen, S.: Spectral pairs in Cartesian coordinates. J. Fourier Anal. Appl. 5(4), 285–302 (1999)
Kolountzakis, M.: Non-symmetric convex domains have no basis of exponentials. Illinois J. Math. 44(3) (1999)
Kolountzakis, M.N.: Multiple lattice tiles and Riesz bases of exponentials. Proc. Am. Math. Soc. 143(2), 741–747 (2015)
Kolountzakis, M., Matolcsi, M.: Complex Hadamard matrices and the spectral set conjecture, Collectanea Mathematica, Supl. pp. 282–291 (2006)
Kolountzakis, M., Matolcsi, M.: Tiles with no spectra. Forum Math. 18(3), 519–528 (2006)
Kozma, G., Nitzan, S.: Combining Riesz bases in \(\mathbb{R}^d\). Rev. Mate. Iberoam. 32(4), 1393–1406 (2016)
Łaba, I.: Fuglede’s conjecture for a union of two intervals. Proc. Am. Math. Soc. 129(10), 2965–2972 (2001)
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)
Marzo, J.: Riesz basis of exponentials for a union of cubes in \({\mathbb{R}}^{d}\). ar**v preprint math/0601288. 2006 Jan 12
Matolcsi, M.: Fuglede conjecture fails in dimension \(4\). Proc. Am. Math. Soc. 133(10), 3021–3026 (2005)
Nitzan, S., Olevskii, A., Ulanovskii, A.: Exponential frames on unbounded sets. Proc. Am. Math. Soc. 144, 109–118 (2016)
Tao, T.: Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11(2–3), 251–258 (2004)
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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