Abstract
Let An ∈ M2 (ℤ) be integral matrices such that the infinite convolution of Dirac measures with equal weights
is a probability measure with compact support, where \(\cal{D}=\{(0,0)^{t},(1,0)^{t},(0,1)^{t}\}\) is the Sierpinski digit. We prove that there exists a set Λ ⊂ ℝ2 such that the family {e2πi〈λ,x〉: λ ∈ Λ} is an orthonormal basis of \(L^{2}(\mu_{\{A_{n},n\geq1\}})\) if and only if \({1\over{3}}(1,-1)A_{n}\in\mathbb{Z}^{2}\) for n ≥ 2 under some metric conditions on An.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
An, L. X., Fu X. Y., Lai C. K.: On spectral Cantor-Moran measures and a variant of Bourgains sum of sine problem. Adv. Math., 349, 84–124 (2019)
An, L. X., He, X. G., Tao, L.: Spectrality of the planar Sierpinski family. J. Math. Anal. Appl., 432, 725–732 (2015)
An, L. X., He, X. G.: A class of spectral Moran measures. J. Funct. Anal., 266, 343–354 (2014)
An, L. X., He, X. G., Lau, K.-S.: Spectrality of a class of infinite convolutions. Adv. Math., 283, 362–376 (2015)
An, L. X., He, X. G., Li, H. X.: Spectrality of infinite Bernoulli convolutions. J. Funct. Anal., 269, 1571–1590 (2015)
Dai, X. R.: When does a Bernoulli convolution admit a spectrum? Adv. Math., 231, 1681–1693 (2012)
Dai, X. R.: Spectra of Cantor measures. Math. Ann., 366, 1621–1647 (2016)
Dai, X. R., Fu, X. Y., Yan, Z. H.: Spectrality of self-affine Sierpinski-type measures on ℝ2. Appl. Comput. Harmon. Anal., 52, 63–81 (2021)
Dai, X. R., He, X. G., Lai, C. K.: Spectral property of Cantor measures with consecutive digits. Adv. Math., 242, 187–208 (2013)
Dai, X. R., He, X. G., Lau, K.-S.: On spectral N-Bernoulli measures. Adv. Math., 259, 511–531 (2014)
Deng, Q. R.: On the spectra of Sierpinski-type self-affine measures. J. Funct. Anal., 270, 4426–4442 (2016)
Deng, Q. R., Chen, J. B.: Uniformity of spectral self-affine measures. Adv. Math., 380, 107568 (2021)
Deng, Q. R., Dong, X. H., Li, M. T.: Tree structure of spectra of spectral self-affine measures. J. Funct. Anal., 277, 937–957 (2019)
Deng, Q. R., Lau, K.-S.: Sierpinski-type spectral self-similar measures. J. Funct. Anal., 269, 1310–1326 (2015)
Dutkay, D., Han, D., Sun, Q.: On spectra of a Cantor measure. Adv. Math., 221, 251–276 (2009)
Dutkay, D., Han, D., Sun, Q.: Divergence of the mock and scrambled Fourier series on fractal measures. Trans, Amer. Math. Soc., 366, 2191–2208 (2014)
Dutkay, D., Han, D., Sun, Q., Weber, E.: On the Beurling dimension of exponential frames. Adv. Math., 226, 285–297 (2011)
Dutkay, D., Hausserman, J., Lai, C. K.: Hadamard triples generate self-affine spectral measures. Trans. Amer. Math. Soc., 371, 1439–1481 (2019)
Dutkay, D., Lai, C. K.: Uniformity of measures with Fourier frames. Adv. Math., 252, 684–707 (2014)
Dutkay, D., Lai, C. K.: Spectral measures generated by arbitrary and random convolutions. J. Math. Pures Appl. (9), 107, 183–204 (2017)
Dutkay, D., Jorgenson, P.: Wavelets on fractals. Rev. Mat. Iberoam., 22, 131–180 (2006)
Dutkay, D., Jorgenson, P.: Analysis of orthogonality and of orbits in affine iterated function systems. Math. Z., 256, 801–823 (2007)
Fu, Y. S., He, X. G., Wen, Z. X.: Spectra of Bernoulli convolutions and random convolutions. J. Math. Pures Appl. (9), 116, 105–131 (2018)
Hu, T. Y., Lau, K.-S.: Spectral property of the Bernoulli convolutions. Adv. Math., 219, 554–567 (2008)
He, X. G., Lai, C. K., Lau, K.-S.: Exponential spectra in L2(μ). Appl. Comput. Harmon. Anal., 34, 327–338 (2013)
Jorgenson, P., Pederson, S.: Dense analytic subspaces in fractal L2-spaces. J. Anal. Math., 75, 185–228 (1998)
Łaba, I.: Fuglede’s conjecture for a union of two intervals. Proc. Amer. Math. Soc., 129(10), 2965–2972 (2001)
Łaba, I., Wang, Y.: On spectral Cantor measures. J. Funct. Anal., 193, 409–420 (2002)
Li, J. L.: Spectral self-affine measures on the planar Sierpinski family. Sci. China Math., 56, 1619–1628 (2013)
Li, J. L.: Non-spectral problem for a class of planar self-affine measures. J. Funct. Anal., 255, 3125–3148 (2008)
Li, J. L.: Orthogonal exponentials on the generalized plane Sierpinski gasket. J. Approx. Theory, 153, 161–169 (2008)
Li, J. L.: Spectrality of self-affine measures on the three-dimensional Sierpinski gasket. Proc. Edinb. Math. Soc., 55, 477–496 (2012)
Liu, J. C., Dong, X. H., Li, J. L.: Non-spectral problem for the planar self-affine measures. J. Funct. Anal., 273, 705–720 (2017)
Strichartz, R.: Convergence of mock Fourier series. J. D’Analyse Math., 99, 333–353 (2006)
Wang, Z. Y., Dong, X.-H.: Spectrality of Sierpinski-Moran measures. Monatsh. Math., 195, 743–761 (2021)
Zhang, M.-M.: Spectrality of Moran Sierpinski-type measures on ℝ2. Canad. Math. Bull., 64, 1024–1040 (2021)
Acknowledgements
Part of the work was done while the corresponding author was visiting the South China Research Center for Applied Mathematics and Interdisciplinary Studies. The author would like to thank Professor Shi-Jin Ding and the South China Research Center for the Applied Mathematics and Interdisciplinary Studies for their hospitality.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
This work is supported by the National Natural Science Foundation of China (Grant Nos. 12371087, 11971109, 11971194, 11672074 and 12271185); the first author is also supported by the program for Probability and Statistics: Theory and Application (Grant No. IRTL1704) and the program for Innovative Research Team in Science and Technology in Fujian Province University (Grant No. IRTSTFJ); the fourth author is also supported by Guangdong NSFC (Grant No. 2022A1515011124)
Rights and permissions
About this article
Cite this article
Deng, Q.R., He, X.G., Li, M.T. et al. The Orthogonal Bases of Exponential Functions Based on Moran-Sierpinski Measures. Acta. Math. Sin.-English Ser. 40, 1804–1824 (2024). https://doi.org/10.1007/s10114-024-2604-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-024-2604-5
Keywords
- Moran-Sierpinski measures
- orthonormal basis of exponential functions
- self-affine measures
- spectral measures