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Spectrality of self-similar measures with product-form digits

  • Juan Su , Sha Wu and Ming-Liang Chen EMAIL logo
Published/Copyright: June 29, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 5

Abstract

A Borel probability measure μ on ℝ is called a spectral measure if it has an exponential orthonormal basis for L 2 ⁢ ( μ ) . In this paper, we consider the spectrality of the self-similar measure μ ρ , D generated by 0 < ρ < 1 and the product-form digit set

D = N p 0 ⁢ L 0 ⁢ { 0 , 1 , … , N - 1 } ⊕ N p 1 ⁢ L 1 ⁢ { 0 , 1 , … , N - 1 } ⊕ ⋯ ⊕ N p s ⁢ L s ⁢ { 0 , 1 , … , N - 1 } ,

where N ≥ 2 is a prime number, 0 ≤ p 0 < p 1 < ⋯ < p s and gcd ⁡ ( L i , N ) = 1 for all 0 ≤ i ≤ s . We show that μ ρ , D is a spectral measure if and only if ρ - 1 = N q ⁢ M and p i ≠ p j ( mod q ) for all 0 ≤ i ≠ j ≤ s , where gcd ⁡ ( M , N ) = 1 .

Keywords: Iterated function system; self-similar measure; spectral measure; product-form digit set
MSC 2010: 28A25; 28A80; 42C05; 46C05

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071125

Award Identifier / Grant number: 11971500

Award Identifier / Grant number: 12171055

Funding source: Natural Science Foundation of Hunan Province

Award Identifier / Grant number: 2019JJ20012

Funding statement: This research is supported in part by the NNSF of China (Grant nos. 12071125, 11971500 and 12171055) and the Hunan Provincial NSF (Grant no. 2019JJ20012).

Acknowledgements

The authors would like to thank the referee for his/her many valuable suggestions.

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Received: 2022-02-27
Revised: 2022-03-31
Published Online: 2022-06-29
Published in Print: 2022-09-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Recognizing pro-𝖱 closures of regular languages
  3. Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
  4. Non-perturbative graph languages, halting problem and complexity
  5. Spectral synthesis via moment functions on hypergroups
  6. A function field analogue of the Stark conjecture
  7. Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
  8. Krull dimension for C*-algebras
  9. The sharp bound of the third Hankel determinant for starlike functions
  10. The role of gentle algebras in higher homological algebra
  11. Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
  12. L p boundedness for a maximal singular integral operator
  13. Common reducing subspaces and decompositions of contractions
  14. A restriction estimate on Morrey spaces
  15. On ideals in group algebras: An uncertainty principle and the Schur product
  16. Spectrality of self-similar measures with product-form digits
  17. Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
  18. On the L q spectra of in-homogeneous self-similar measures
https://doi.org/10.1515/forum-2022-0069
Keywords for this article
Iterated function system; self-similar measure; spectral measure; product-form digit set

Articles in the same Issue

  1. Frontmatter
  2. Recognizing pro-𝖱 closures of regular languages
  3. Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
  4. Non-perturbative graph languages, halting problem and complexity
  5. Spectral synthesis via moment functions on hypergroups
  6. A function field analogue of the Stark conjecture
  7. Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
  8. Krull dimension for C*-algebras
  9. The sharp bound of the third Hankel determinant for starlike functions
  10. The role of gentle algebras in higher homological algebra
  11. Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
  12. L p boundedness for a maximal singular integral operator
  13. Common reducing subspaces and decompositions of contractions
  14. A restriction estimate on Morrey spaces
  15. On ideals in group algebras: An uncertainty principle and the Schur product
  16. Spectrality of self-similar measures with product-form digits
  17. Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
  18. On the L q spectra of in-homogeneous self-similar measures
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