Spectral Eigen-subspace and Tree Structure for a Cantor Measure
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| Format: | Preprint |
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2024
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| author | Deng, Guotai Fu, Yan-Song Kang, Qingcan |
| author_facet | Deng, Guotai Fu, Yan-Song Kang, Qingcan |
| contents | In this work we investigate the question of constructions of the possible Fourier bases $E(Λ)=\{e^{2πi λx}:λ\inΛ\}$ for the Hilbert space $L^2(μ_4)$, where $μ_4$ is the standard middle-fourth Cantor measure and $Λ$ is a countable discrete set. We show that the set $$\mathop \bigcap_{p\in 2\Z+1}\left\{Λ\subset \R: \text{$E(Λ)$ and $E(pΛ)$ are Fourier bases for $L^2(μ_4)$}\right\}$$ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2(μ_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $Λ$ such that $E(Λ)$ and its all odd scaling sets $E(Λ),p\in2\Z+1,$ are still Fourier bases for $L^2(μ_4)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Eigen-subspace and Tree Structure for a Cantor Measure Deng, Guotai Fu, Yan-Song Kang, Qingcan Functional Analysis Primary 42A10, 28A80, 42A65 In this work we investigate the question of constructions of the possible Fourier bases $E(Λ)=\{e^{2πi λx}:λ\inΛ\}$ for the Hilbert space $L^2(μ_4)$, where $μ_4$ is the standard middle-fourth Cantor measure and $Λ$ is a countable discrete set. We show that the set $$\mathop \bigcap_{p\in 2\Z+1}\left\{Λ\subset \R: \text{$E(Λ)$ and $E(pΛ)$ are Fourier bases for $L^2(μ_4)$}\right\}$$ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2(μ_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $Λ$ such that $E(Λ)$ and its all odd scaling sets $E(Λ),p\in2\Z+1,$ are still Fourier bases for $L^2(μ_4)$. |
| title | Spectral Eigen-subspace and Tree Structure for a Cantor Measure |
| topic | Functional Analysis Primary 42A10, 28A80, 42A65 |
| url | https://arxiv.org/abs/2407.13075 |