Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929206139551744 |
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| author | Berner, Chad |
| author_facet | Berner, Chad |
| contents | It is known that if a finite Borel measure $μ$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(μ)$, then $μ$ is of pure type. In this paper, we prove the existence of a class of finite Borel measures $μ$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(μ)$. We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(μ)$ where $μ$ is a singular Borel probability measure on $\mathbb{R}$. Finally, we show measures $μ$ on $[0,1)$ that possess these frame-like Fourier expansions for $L^{2}(μ)$ have all $f\in L^{2}(μ)$ as $L^{2}(μ)$ limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$ Berner, Chad Functional Analysis 42A16, 42A20, 42C15(Primary), 46C07 (Secondary) It is known that if a finite Borel measure $μ$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(μ)$, then $μ$ is of pure type. In this paper, we prove the existence of a class of finite Borel measures $μ$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(μ)$. We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(μ)$ where $μ$ is a singular Borel probability measure on $\mathbb{R}$. Finally, we show measures $μ$ on $[0,1)$ that possess these frame-like Fourier expansions for $L^{2}(μ)$ have all $f\in L^{2}(μ)$ as $L^{2}(μ)$ limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk. |
| title | Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$ |
| topic | Functional Analysis 42A16, 42A20, 42C15(Primary), 46C07 (Secondary) |
| url | https://arxiv.org/abs/2401.05243 |