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Beurling Dimension And Self-Affine Measures

Author

Listed:
  • MIN-WEI TANG

    (Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), (Ministry of Education of China), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, P. R. China)

  • ZHI-YI WU

    (Department of Mathematical Sciences, University of Oulu, P. O. Box 3000, 90014 Oulu, Finland)

Abstract

In this paper, we continue the study in our previous work Beurling dimenison and self-similar measures [J. Funct. Anal. 274 (2018) 2245–2264]. We analyze the Beurling dimension of Bessel sequences and frame spectra of self-affine measures on ℠d. Unlike the case for the self-similar measures, it is very difficult to obtain a general expression for the dimension of self-affine sets. This implies that the Hausdorff dimension may be not a good candidate that controls the Beurling dimension in general. Instead, we find that the pseudo Hausdorff dimension proposed by He and Lau [Math. Nachr. 281 (2008) 1142–1158] can be a good candidate that can control the Beurling dimension. With the help of the pseudo Hausdorff dimension, we obtain the upper bounds of the Beurling dimension of Bessel sequences. Under suitable condition, we give the lower bound of the Beurling dimension of frame spectra of self-affine measures on ℠d. Some examples are given to illustrate our theory.

Suggested Citation

  • Min-Wei Tang & Zhi-Yi Wu, 2021. "Beurling Dimension And Self-Affine Measures," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-12, September.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:06:n:s0218348x21501747
    DOI: 10.1142/S0218348X21501747
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