Comptes Rendus
Harmonic analysis
The spectrality of symmetric additive measures
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 783-793.

Let ρ be a symmetric measure of Lebesgue type, i.e.,

ρ=1 2(μ×δ 0 +δ 0 ×μ),

where the component measure μ is the Lebesgue measure supported on [t,t+1] for t{-1 2} and δ 0 is the Dirac measure at 0. We prove that ρ is a spectral measure if and only if t1 2. In this case, L 2 (ρ) has a unique orthonormal basis of the form

e 2πi(λx-λy) :λΛ 0 ,

where Λ 0 is the spectrum of the Lebesgue measure supported on [-t-1,-t][t,t+1]. Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021].

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.435
Classification: 28A80, 42C05

Wen-Hui Ai 1; Zheng-Yi Lu 1; Ting Zhou 2

1 Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
2 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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},
}
TY  - JOUR
AU  - Wen-Hui Ai
AU  - Zheng-Yi Lu
AU  - Ting Zhou
TI  - The spectrality of symmetric additive measures
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 783
EP  - 793
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.435
LA  - en
ID  - CRMATH_2023__361_G4_783_0
ER  - 
%0 Journal Article
%A Wen-Hui Ai
%A Zheng-Yi Lu
%A Ting Zhou
%T The spectrality of symmetric additive measures
%J Comptes Rendus. Mathématique
%D 2023
%P 783-793
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.435
%G en
%F CRMATH_2023__361_G4_783_0
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] Ole Christensen An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser, 2003 | DOI | Zbl

[2] Xin-Rong Dai When does a Bernoulli convolution admit a spectrum?, Adv. Math., Volume 231 (2012) no. 3-4, pp. 1681-1693 | MR | Zbl

[3] Xin-Rong Dai; Xiao-Ye Fu; Zhi-Hui Yan Spectrality of self-affine Sierpinski-type measures on 2 , Appl. Comput. Harmon. Anal., Volume 52 (2021), pp. 63-81 | MR | Zbl

[4] Xin-Rong Dai; Xing-Gang He; Ka-Sing Lau On spectral N-Bernoulli measures, Adv. Math., Volume 259 (2014), pp. 511-531 | MR | Zbl

[5] Alberto Debernardi; Nir Lev 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] Dorin Ervin Dutkay; Chun-Kit Lai Uniformity of measures with Fourier frames, Adv. Math., Volume 252 (2014), pp. 684-707 | DOI | MR | Zbl

[7] Bent Fuglede 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] Palle E. T. Jørgensen; Steen Pedersen Dense analytic subspaces in fractal L 2 -spaces, J. Anal. Math., Volume 75 (1998), pp. 185-228 | DOI | MR | Zbl

[9] Mihail N. Kolountzakis; Máté Matolcsi Tiles with no spectra, Forum Math., Volume 18 (2006) no. 3, pp. 519-528 | MR | Zbl

[10] Chun-Kit Lai; Bochen Liu; Hal Prince Spectral properties of some unions of linear spaces, J. Funct. Anal., Volume 280 (2021) no. 11, 108985, 32 pages | MR | Zbl

[11] Henry J. Landau Necessary density conditions for sampling an interpolation of certain entire functions, Acta Math., Volume 117 (1967), pp. 37-52 | DOI | MR | Zbl

[12] Nir Lev Fourier frames for singular measures and pure type phenomena, Proc. Am. Math. Soc., Volume 146 (2018) no. 7, pp. 2883-2896 | MR | Zbl

[13] Máté Matolcsi Fuglede conjecture fails in dimension 4, Proc. Am. Math. Soc., Volume 133 (2005) no. 10, pp. 3021-3026 | DOI | MR | Zbl

[14] Joaquim Ortega-Cerdà; Kristian Seip Fourier frames, Ann. Math., Volume 155 (2002) no. 3, pp. 789-806 | DOI | MR | Zbl

[15] Terence Tao Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett., Volume 11 (2004) no. 2-3, pp. 251-258 | MR | Zbl

Cited by Sources:

Comments - Policy