Shift-invariant subspaces of Sobolev type
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911891112067072 |
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| author | Aksentijević, Aleksandar Aleksić, Suzana |
| author_facet | Aksentijević, Aleksandar Aleksić, Suzana |
| contents | This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\in\mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s(\mathbb{R}^n)$, are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier $\left(1-\fracΔ{4π^2}\right)^{s/2}f=\mathcal{F}^{-1}\big((1+|t|^2)^{s/2}\widehat{f}(t)\big)$, we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shift-invariant subspaces of Sobolev type Aksentijević, Aleksandar Aleksić, Suzana Functional Analysis 42C15, 42B30 This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\in\mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s(\mathbb{R}^n)$, are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier $\left(1-\fracΔ{4π^2}\right)^{s/2}f=\mathcal{F}^{-1}\big((1+|t|^2)^{s/2}\widehat{f}(t)\big)$, we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type. |
| title | Shift-invariant subspaces of Sobolev type |
| topic | Functional Analysis 42C15, 42B30 |
| url | https://arxiv.org/abs/2405.17943 |