Extending wavelet regularity beyond Gevrey classes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911423675760640 |
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| author | Tomić, Filip Tutić, Stefan Žigić, Milica |
| author_facet | Tomić, Filip Tutić, Stefan Žigić, Milica |
| contents | We construct a smooth orthonormal wavelet $ψ$ such that both $ψ$ and its Fourier transform $\widehatψ$ belong to the extended Gevrey class $\mathcal{E}_σ(\mathbb{R})$ for $σ> 1$, providing an example that lies beyond all classical Gevrey classes. Our approach uses the idea of invariant cycles to extend the initial Lemarié-Meyer support of the low-pass filter $m_0$ from $ [-\frac{2π}{3}, \frac{2π}{3}]$ to $ [-\frac{4π}{5}, \frac{4π}{5}]$. This extension allows us to control the decay rate of $m_0$ near $\frac{2π}{3}$, which yields global decay estimates for $ψ$ and $\hatψ$. In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending wavelet regularity beyond Gevrey classes Tomić, Filip Tutić, Stefan Žigić, Milica Functional Analysis We construct a smooth orthonormal wavelet $ψ$ such that both $ψ$ and its Fourier transform $\widehatψ$ belong to the extended Gevrey class $\mathcal{E}_σ(\mathbb{R})$ for $σ> 1$, providing an example that lies beyond all classical Gevrey classes. Our approach uses the idea of invariant cycles to extend the initial Lemarié-Meyer support of the low-pass filter $m_0$ from $ [-\frac{2π}{3}, \frac{2π}{3}]$ to $ [-\frac{4π}{5}, \frac{4π}{5}]$. This extension allows us to control the decay rate of $m_0$ near $\frac{2π}{3}$, which yields global decay estimates for $ψ$ and $\hatψ$. In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction. |
| title | Extending wavelet regularity beyond Gevrey classes |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2512.05655 |