Spectrality of a measure consisting of two line segments
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910802194202624 |
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| author | Kolountzakis, Mihail N. Wu, Sha |
| author_facet | Kolountzakis, Mihail N. Wu, Sha |
| contents | Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $ρ$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of $t$ (being spectral means that there is an orthonormal basis for $L^2(ρ)$ consisting of exponentials $e^{2πi (λ_1 x + λ_2 y)}$). We almost complete the study showing that for $-\frac12<t<0$ and for all $t \notin {\mathbb Q}$ the measure $ρ$ is not spectral. The only remaining undecided case is the case $t=-\frac12$ (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectrality of a measure consisting of two line segments Kolountzakis, Mihail N. Wu, Sha Classical Analysis and ODEs Functional Analysis 42C15, 42C30 Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $ρ$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of $t$ (being spectral means that there is an orthonormal basis for $L^2(ρ)$ consisting of exponentials $e^{2πi (λ_1 x + λ_2 y)}$). We almost complete the study showing that for $-\frac12<t<0$ and for all $t \notin {\mathbb Q}$ the measure $ρ$ is not spectral. The only remaining undecided case is the case $t=-\frac12$ (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum. |
| title | Spectrality of a measure consisting of two line segments |
| topic | Classical Analysis and ODEs Functional Analysis 42C15, 42C30 |
| url | https://arxiv.org/abs/2501.11367 |