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On the Fourier orthonormal bases of a class of self-similar measures on ℝ n

  • Wei Tang and Zhi-Yong Wang EMAIL logo
Published/Copyright: July 1, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 35 Issue 6

Abstract

Let μ M , D be a self-similar measure generated by an n × n expanding real matrix M = ρ - 1 ⁢ I and a finite digit set D ⊂ ℤ n , where 0 < | ρ | < 1 and I is an n × n unit matrix. In this paper, we study the existence of a Fourier basis for L 2 ⁢ ( μ M , D ) , i.e., we find a discrete set Λ such that E Λ = { e 2 ⁢ π ⁢ i ⁢ 〈 λ , x 〉 : λ ∈ Λ } is an orthonormal basis for L 2 ⁢ ( μ M , D ) . Under some suitable conditions for D, some necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) to admit infinite orthogonal exponential functions are given. Then we set up a framework to obtain necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) to have a Fourier basis. Finally, we demonstrate how these results can be applied to self-similar measures.

Keywords: Iterated function system; self-affine measure; spectral measure; orthogonal
MSC 2020: 28A80; 42C05

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11901187

Award Identifier / Grant number: 12001183

Award Identifier / Grant number: 11831007

Funding source: Natural Science Foundation of Hunan Province

Award Identifier / Grant number: 2020JJ5097

Funding statement: The research is supported in part by the NNSF of China (Nos. 11901187, 12001183 and 11831007) and the NSF of Hunan Province (No. 2020JJ5097).

Acknowledgements

The authors are grateful to the anonymous referee for some valuable suggestions and comments.

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Received: 2023-01-11
Revised: 2023-05-21
Published Online: 2023-07-01
Published in Print: 2023-11-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields
  3. Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
  4. Minimal Kähler submanifolds in product of space forms
  5. On the number of rational points of certain algebraic varieties over finite fields
  6. On two conjectures of Sun concerning Apéry-like series
  7. A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth
  8. Time-step heat problem on the mesh: asymptotic behavior and decay rates
  9. Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras
  10. Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
  11. Skew-braces and 𝑞-braces
  12. Products of unipotent elements in certain algebras
  13. On the Fourier orthonormal bases of a class of self-similar measures on ℝ n
  14. The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
  15. Restriction estimates in a conical singular space: Schrödinger equation
  16. Fractional integrals associated with Radon transforms
https://doi.org/10.1515/forum-2023-0006
Keywords for this article
Iterated function system; self-affine measure; spectral measure; orthogonal

Articles in the same Issue

  1. Frontmatter
  2. Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields
  3. Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
  4. Minimal Kähler submanifolds in product of space forms
  5. On the number of rational points of certain algebraic varieties over finite fields
  6. On two conjectures of Sun concerning Apéry-like series
  7. A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth
  8. Time-step heat problem on the mesh: asymptotic behavior and decay rates
  9. Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras
  10. Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
  11. Skew-braces and 𝑞-braces
  12. Products of unipotent elements in certain algebras
  13. On the Fourier orthonormal bases of a class of self-similar measures on ℝ n
  14. The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
  15. Restriction estimates in a conical singular space: Schrödinger equation
  16. Fractional integrals associated with Radon transforms
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