Let be a symmetric measure of Lebesgue type, i.e.,
where the component measure is the Lebesgue measure supported on for and is the Dirac measure at . We prove that is a spectral measure if and only if . In this case, has a unique orthonormal basis of the form
where is the spectrum of the Lebesgue measure supported on . Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021].
Accepted:
Published online:
Wen-Hui Ai 1; Zheng-Yi Lu 1; Ting Zhou 2
@article{CRMATH_2023__361_G4_783_0, author = {Wen-Hui Ai and Zheng-Yi Lu and Ting Zhou}, title = {The spectrality of symmetric additive measures}, journal = {Comptes Rendus. Math\'ematique}, pages = {783--793}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.435}, language = {en}, }
Wen-Hui Ai; Zheng-Yi Lu; Ting Zhou. The spectrality of symmetric additive measures. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 783-793. doi : 10.5802/crmath.435. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.435/
[1] An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser, 2003 | DOI | Zbl
[2] When does a Bernoulli convolution admit a spectrum?, Adv. Math., Volume 231 (2012) no. 3-4, pp. 1681-1693 | MR | Zbl
[3] Spectrality of self-affine Sierpinski-type measures on , Appl. Comput. Harmon. Anal., Volume 52 (2021), pp. 63-81 | MR | Zbl
[4] On spectral N-Bernoulli measures, Adv. Math., Volume 259 (2014), pp. 511-531 | MR | Zbl
[5] Riesz bases of exponentials for convex polytopes with symmetric faces, J. Eur. Math. Soc., Volume 24 (2022) no. 8, pp. 3017-3029 | DOI | MR | Zbl
[6] Uniformity of measures with Fourier frames, Adv. Math., Volume 252 (2014), pp. 684-707 | DOI | MR | Zbl
[7] Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal., Volume 16 (1974) no. 1, pp. 101-121 | DOI | MR | Zbl
[8] Dense analytic subspaces in fractal -spaces, J. Anal. Math., Volume 75 (1998), pp. 185-228 | DOI | MR | Zbl
[9] Tiles with no spectra, Forum Math., Volume 18 (2006) no. 3, pp. 519-528 | MR | Zbl
[10] Spectral properties of some unions of linear spaces, J. Funct. Anal., Volume 280 (2021) no. 11, 108985, 32 pages | MR | Zbl
[11] Necessary density conditions for sampling an interpolation of certain entire functions, Acta Math., Volume 117 (1967), pp. 37-52 | DOI | MR | Zbl
[12] Fourier frames for singular measures and pure type phenomena, Proc. Am. Math. Soc., Volume 146 (2018) no. 7, pp. 2883-2896 | MR | Zbl
[13] Fuglede conjecture fails in dimension , Proc. Am. Math. Soc., Volume 133 (2005) no. 10, pp. 3021-3026 | DOI | MR | Zbl
[14] Fourier frames, Ann. Math., Volume 155 (2002) no. 3, pp. 789-806 | DOI | MR | Zbl
[15] Fuglede’s conjecture is false in and higher dimensions, Math. Res. Lett., Volume 11 (2004) no. 2-3, pp. 251-258 | MR | Zbl
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