Spectral property of the planar self-affine measures with three-element digit sets

Ming-Liang Chen 2 , Jing-Cheng Liu 3  and Juan Su 1
  • 1 School of Mathematics and Statistics, Changsha University of Science & Technology, Hunan 410114, Changsha, P. R. China
  • 2 School of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. China
  • 3 Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, College of Mathematics and Statistics, Changsha, P. R. China
Ming-Liang Chen, Jing-Cheng Liu
  • Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
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and Juan Su
  • Corresponding author
  • School of Mathematics and Statistics, Changsha University of Science & Technology, Changsha, Hunan 410114, P. R. China
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Abstract

Let the self-affine measure μM,D be generated by an expanding real matrix M=diag(ρ1-1,ρ2-1) and an integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t} with α1β2-α2β10. In this paper, the sufficient and necessary conditions for L2(μM,D) to contain an infinite orthogonal set of exponential functions are given.

  • [1]

    L.-X. An, X.-G. He and K.-S. Lau, Spectrality of a class of infinite convolutions, Adv. Math. 283 (2015), 362–376.

    • Crossref
    • Export Citation
  • [2]

    X.-R. Dai, When does a Bernoulli convolution admit a spectrum?, Adv. Math. 231 (2012), no. 3–4, 1681–1693.

    • Crossref
    • Export Citation
  • [3]

    X.-R. Dai, X.-G. He and C.-K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013), 187–208.

    • Crossref
    • Export Citation
  • [4]

    X.-R. Dai, X.-G. He and K.-S. Lau, On spectral N-Bernoulli measures, Adv. Math. 259 (2014), 511–531.

    • Crossref
    • Export Citation
  • [5]

    Q.-R. Deng, Spectrality of one dimensional self-similar measures with consecutive digits, J. Math. Anal. Appl. 409 (2014), no. 1, 331–346.

    • Crossref
    • Export Citation
  • [6]

    Q.-R. Deng, On the spectra of Sierpinski-type self-affine measures, J. Funct. Anal. 270 (2016), no. 12, 4426–4442.

    • Crossref
    • Export Citation
  • [7]

    Q.-R. Deng and K.-S. Lau, Sierpinski-type spectral self-similar measures, J. Funct. Anal. 269 (2015), no. 5, 1310–1326.

    • Crossref
    • Export Citation
  • [8]

    D. E. Dutkay, J. Haussermann and C.-K. Lai, Hadamard triples generate self-affine spectral measures, Trans. Amer. Math. Soc. 371 (2019), no. 2, 1439–1481.

  • [9]

    D. E. Dutkay and P. E. T. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007), no. 4, 801–823.

    • Crossref
    • Export Citation
  • [10]

    D. E. Dutkay and P. E. T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (2007), no. 1, 110–137.

    • Crossref
    • Export Citation
  • [11]

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

    • Crossref
    • Export Citation
  • [12]

    T.-Y. Hu and K.-S. Lau, Spectral property of the Bernoulli convolutions, Adv. Math. 219 (2008), no. 2, 554–567.

    • Crossref
    • Export Citation
  • [13]

    J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.

    • Crossref
    • Export Citation
  • [14]

    P. E. T. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal L 2 L^{2}-spaces, J. Anal. Math. 75 (1998), 185–228.

    • Crossref
    • Export Citation
  • [15]

    M. N. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. (2006), no. Vol. Extra, 281–291.

  • [16]

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

  • [17]

    I. Ł aba and Y. Wang, On spectral Cantor measures, J. Funct. Anal. 193 (2002), no. 2, 409–420.

    • Crossref
    • Export Citation
  • [18]

    J.-L. Li, Non-spectral problem for a class of planar self-affine measures, J. Funct. Anal. 255 (2008), no. 11, 3125–3148.

    • Crossref
    • Export Citation
  • [19]

    J.-L. Li, Spectra of a class of self-affine measures, J. Funct. Anal. 260 (2011), no. 4, 1086–1095.

    • Crossref
    • Export Citation
  • [20]

    J.-L. Li, Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl. 432 (2015), no. 2, 1005–1017.

    • Crossref
    • Export Citation
  • [21]

    J.-C. Liu, X.-H. Dong and J.-L. Li, Non-spectral problem for the planar self-affine measures, J. Funct. Anal. 273 (2017), no. 2, 705–720.

    • Crossref
    • Export Citation
  • [22]

    Z.-Y. Lu, X.-H. Dong and P.-F. Zhang, Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket, Forum Math. 31 (2019), no. 6, 1447–1455.

    • Crossref
    • Export Citation
  • [23]

    M. B. Nathanson, Elementary Methods in Number Theory, Grad. Texts in Math. 195, Springer, New York, 2000.

  • [24]

    F. P. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc. (2) 30 (1929), no. 4, 264–286.

  • [25]

    T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2–3, 251–258.

    • Crossref
    • Export Citation
  • [26]

    Z.-Y. Wang, Z.-M. Wang, X.-H. Dong and P.-F. Zhang, Orthogonal exponential functions of self-similar measures with consecutive digits in \mathbb{R}, J. Math. Anal. Appl. 467 (2018), no. 2, 1148–1152.

    • Crossref
    • Export Citation
  • [27]

    Z. H. Yan, Spectrality of certain fractal measures on n {\mathbb{R}^{n}}, PhD. thesis, Sun Yat-sen University, 2019.

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