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The exact number of orthogonal exponentials on the spatial Sierpinski gasket

  • Qi Wang EMAIL logo
Published/Copyright: August 26, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 5

Abstract

Let μM,D be a self-affine measure associated with an expanding real matrix M=diag⁡[ρ1,ρ2,ρ3] and the digit set D={0,e1,e2,e3} in the space ℝ3, where |ρ1|,|ρ2|,|ρ3|∈(1,∞) and e1,e2,e3 is the standard basis of unit column vectors in ℝ3. In this paper, we mainly consider the case

ρ1∈{pq:p∈2⁢ℤ,q∈2⁢ℤ-1},ρ2,ρ3∈{pq:p,q∈2⁢ℤ-1}.

We prove that if ρ2=ρ3, then there exist at most 4 mutually orthogonal exponential functions in the Hilbert space L2⁢(μM,D), where the number 4 is the best upper bound. If ρ2=-ρ3, then there exist at most 8 mutually orthogonal exponential functions in L2⁢(μM,D), where the number 8 is the best upper bound. If |ρ3|≠|ρ2|, then there are any number of orthogonal exponentials in L2⁢(μM,D). This gives the exact number of orthogonal exponentials on the spatial Sierpinski gasket in the above case.

Keywords: Self-affine measures; Sierpinski gasket; orthogonal exponentials; non-spectrality
MSC 2010: 28A80; 42C05; 46C05

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12001346

Funding statement: This work is supported by the National Natural Science Foundation of China (No. 12001346) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2020JQ-695). This is also supported in part by an innovation team on computationally efficient numerical methods based on new energy problems in Shaanxi province, and an innovative team project of Shaanxi Provincial Department of Education (No. 21JP013).

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Received: 2021-02-24
Revised: 2021-06-08
Published Online: 2021-08-26
Published in Print: 2021-09-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group
  3. Extrapolation to product Herz spaces and some applications
  4. The exact number of orthogonal exponentials on the spatial Sierpinski gasket
  5. A note on special cubic fourfolds of small discriminants
  6. Strong multiplicity one for Siegel cusp forms of degree two
  7. Gross–Prasad periods for reducible representations
  8. LS-category of moment-angle manifolds and higher order Massey products
  9. Estimates of heat kernels of non-symmetric Lévy processes
  10. Global continuity of variational solutions weakening the one-sided bounded slope condition
  11. Finitary birepresentations of finitary bicategories
  12. Solvability for nonlocal boundary value problems with generalized 𝑝-Laplacian on an unbounded domain
  13. Conformal Killing forms on 2-step nilpotent Riemannian Lie groups
  14. Simply transitive NIL-affine actions of solvable Lie groups
  15. Mahler measure of 3D Landau–Ginzburg potentials
https://doi.org/10.1515/forum-2021-0050
Keywords for this article
Self-affine measures; Sierpinski gasket; orthogonal exponentials; non-spectrality

Articles in the same Issue

  1. Frontmatter
  2. Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group
  3. Extrapolation to product Herz spaces and some applications
  4. The exact number of orthogonal exponentials on the spatial Sierpinski gasket
  5. A note on special cubic fourfolds of small discriminants
  6. Strong multiplicity one for Siegel cusp forms of degree two
  7. Gross–Prasad periods for reducible representations
  8. LS-category of moment-angle manifolds and higher order Massey products
  9. Estimates of heat kernels of non-symmetric Lévy processes
  10. Global continuity of variational solutions weakening the one-sided bounded slope condition
  11. Finitary birepresentations of finitary bicategories
  12. Solvability for nonlocal boundary value problems with generalized 𝑝-Laplacian on an unbounded domain
  13. Conformal Killing forms on 2-step nilpotent Riemannian Lie groups
  14. Simply transitive NIL-affine actions of solvable Lie groups
  15. Mahler measure of 3D Landau–Ginzburg potentials
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