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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.00677 |
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| _version_ | 1866909454705885184 |
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| author | Aimar, Hugo Arias, Carlos Exequiel Gómez, Ivana |
| author_facet | Aimar, Hugo Arias, Carlos Exequiel Gómez, Ivana |
| contents | We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $α$ regularity of $f$ with respect to the ultrametric $δ(x,y)=\inf \{|I|: x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $α$ is positive. Precisely, $f\in \textrm{Lip}_δ(α)$ if and only if $\left\vert\left<f,h^j_k\right>\right\vert\leq C 2^{-(α+ \tfrac{1}{2})j}$, for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Haar wavelet characterization of dyadic Lipschitz regularity Aimar, Hugo Arias, Carlos Exequiel Gómez, Ivana Classical Analysis and ODEs 42C15 We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $α$ regularity of $f$ with respect to the ultrametric $δ(x,y)=\inf \{|I|: x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $α$ is positive. Precisely, $f\in \textrm{Lip}_δ(α)$ if and only if $\left\vert\left<f,h^j_k\right>\right\vert\leq C 2^{-(α+ \tfrac{1}{2})j}$, for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$. |
| title | Haar wavelet characterization of dyadic Lipschitz regularity |
| topic | Classical Analysis and ODEs 42C15 |
| url | https://arxiv.org/abs/2403.00677 |