Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929324695748608 |
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| author | Holighaus, Nicki Koliander, Günther |
| author_facet | Holighaus, Nicki Koliander, Günther |
| contents | We provide an example for the generating matrix $A$ of a two-dimensional lattice $Γ= A\mathbb{Z}^2$, such that the following holds: For any sufficiently smooth and localized mother wavelet $ψ$, there is a constant $β(A,ψ)>0$, such that $βΓ\cap (\mathbb{R}\times\mathbb{R}^+)$ is a set of stable sampling for the wavelet system generated by $ψ$, for all $0<β\leq β(A,ψ)$. The result and choice of the generating matrix are loosely inspired by the studies of low discrepancy sequences and uniform distribution modulo $1$. In particular, we estimate the number of lattice points contained in any axis parallel rectangle of fixed area. This estimate is combined with a recent sampling result for continuous wavelet systems, obtained via the oscillation method of general coorbit theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems Holighaus, Nicki Koliander, Günther Functional Analysis 42C40 (Primary), 42C15, 11K38 (Secondary) We provide an example for the generating matrix $A$ of a two-dimensional lattice $Γ= A\mathbb{Z}^2$, such that the following holds: For any sufficiently smooth and localized mother wavelet $ψ$, there is a constant $β(A,ψ)>0$, such that $βΓ\cap (\mathbb{R}\times\mathbb{R}^+)$ is a set of stable sampling for the wavelet system generated by $ψ$, for all $0<β\leq β(A,ψ)$. The result and choice of the generating matrix are loosely inspired by the studies of low discrepancy sequences and uniform distribution modulo $1$. In particular, we estimate the number of lattice points contained in any axis parallel rectangle of fixed area. This estimate is combined with a recent sampling result for continuous wavelet systems, obtained via the oscillation method of general coorbit theory. |
| title | Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems |
| topic | Functional Analysis 42C40 (Primary), 42C15, 11K38 (Secondary) |
| url | https://arxiv.org/abs/2307.13481 |