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Construction of a class of spectral measures

  • Hai-Hua Wu and Jing-Cheng Liu EMAIL logo
Published/Copyright: August 20, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

Let the iterated function systems { S i } i = 1 N be defined by S i ⁢ ( x ) = ( - 1 ) i - 1 ⁢ ρ ⁢ ( x + d i ) , x ∈ R , d i ∈ D , where 0 < ρ < 1 , and 𝐷 is a finite subset of ℤ. Let the measure μ ρ , P be generated by the IFS and the probability weight P = { p i } i = 1 N . In this paper, we introduce a new way to consider the spectrality of μ ρ , P , and obtain a sufficient and necessary condition for N = 2 , and construct two examples of spectral measures for N = 2 ⁢ L ≥ 4 and N = 2 ⁢ L + 1 , respectively.

Keywords: Fourier transform; spectrum; spectral measure; self-affine measure; compatible pair
MSC 2010: 28A80; 42C05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171055

Award Identifier / Grant number: 12071125

Award Identifier / Grant number: 11831007

Funding source: Natural Science Foundation of Hunan Province

Award Identifier / Grant number: 2021JJ40559

Funding source: Scientific Research Foundation of Hunan Provincial Education Department

Award Identifier / Grant number: 21C0196

Funding statement: The research is partially supported by the NNSF of China (Nos. 12171055, 12071125, 11831007), the NSF of Hunan Province (No. 2021JJ40559), and the SRF of Hunan Provincial Education Department (No. 21C0196).

  1. Communicated by: Siegfried Echterhoff

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Received: 2022-01-11
Revised: 2022-05-24
Published Online: 2022-08-20
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
https://doi.org/10.1515/forum-2022-0011
Keywords for this article
Fourier transform; spectrum; spectral measure; self-affine measure; compatible pair

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
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