The spectral eigenvalues of a class of product-form self-similar spectral measure
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916010492166144 |
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| author | Xiao-yu, Yan Wen-hui, Ai |
| author_facet | Xiao-yu, Yan Wen-hui, Ai |
| contents | Let μ_{M,D} be the self-similar measure generated by the positive integer M=RN^q and the product-form digit set D=\{0,1,\dots,N-1\}\oplus N^{p_1}\{0,1,\dots,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\dots,N-1\}, where R>1, N>1, q, p_i(1\leq i\leq s) are positive integers with gcd(R,N)=1 and p_1<p_2<\cdots<p_s<q. In this paper, we first show that μ_{M,D} is a spectral measure with a model spectrum Λ. Then we completely settle two types of spectral eigenvalue problems for μ_{M,D}. On the first case, for a real t, we give a necessary and sufficient condition under which tΛis also a spectrum of μ_{M,D}. On the second case, we characterize all possible real numbers t such that there exists a countable set Λ'\subset \mathbb{R} such that Λ' and tΛ' are both spectra of μ_{M,D}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08714 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spectral eigenvalues of a class of product-form self-similar spectral measure Xiao-yu, Yan Wen-hui, Ai Functional Analysis 28A80, 42C05 Let μ_{M,D} be the self-similar measure generated by the positive integer M=RN^q and the product-form digit set D=\{0,1,\dots,N-1\}\oplus N^{p_1}\{0,1,\dots,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\dots,N-1\}, where R>1, N>1, q, p_i(1\leq i\leq s) are positive integers with gcd(R,N)=1 and p_1<p_2<\cdots<p_s<q. In this paper, we first show that μ_{M,D} is a spectral measure with a model spectrum Λ. Then we completely settle two types of spectral eigenvalue problems for μ_{M,D}. On the first case, for a real t, we give a necessary and sufficient condition under which tΛis also a spectrum of μ_{M,D}. On the second case, we characterize all possible real numbers t such that there exists a countable set Λ'\subset \mathbb{R} such that Λ' and tΛ' are both spectra of μ_{M,D}. |
| title | The spectral eigenvalues of a class of product-form self-similar spectral measure |
| topic | Functional Analysis 28A80, 42C05 |
| url | https://arxiv.org/abs/2408.08714 |