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THE SPECTRAL EIGENVALUES OF A CLASS OF PRODUCT-FORM SELF-SIMILAR SPECTRAL MEASURE

Part of: Nontrigonometric harmonic analysis Classical measure theory

Published online by Cambridge University Press:  21 July 2025

XIAO-YU YAN  and
WEN-HUI AI Open the ORCID record for WEN-HUI AI [Opens in a new window]
XIAO-YU YAN
Affiliation:
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, https://ror.org/053w1zy07 Hunan Normal University , Changsha, Hunan 410081, PR China e-mail: xyyan1103@163.com
WEN-HUI AI*
Affiliation:
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, https://ror.org/053w1zy07 Hunan Normal University , Changsha, Hunan 410081, PR China
*
e-mail: awhxyz123@163.com
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Abstract

Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where $R\geq 2$, $N\geq 2$, q, $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$.

Keywords

self-similar measurespectral measureproduct-form digit setspectral eigenvalue problems

MSC classification

Primary: 28A80: Fractals
Secondary: 42C05: Orthogonal functions and polynomials, general theory

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 119 , Issue 2 , October 2025 , pp. 246 - 263
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ji Li

The research is supported in part by the NNSF of China (Nos. 12201206 and 12371072) and the Hunan Provincial NSF (No. 2024JJ6301).

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