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arXiv · 2512.05655

Extending wavelet regularity beyond Gevrey classes

Filip Tomić·Stefan Tutić·Milica Žigić
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Abstract

We construct a smooth orthonormal wavelet $\psi$ such that both $\psi$ and its Fourier transform $\widehat{\psi}$ belong to the extended Gevrey class $\mathcal{E}_{\sigma}(\mathbb{R})$ for $\sigma > 1$, providing an example that lies beyond all classical Gevrey classes. Our approach uses the idea of invariant cycles to extend the initial Lemari\'e-Meyer support of the low-pass filter $m_0$ from $ [-\frac{2\pi}{3}, \frac{2\pi}{3}]$ to $ [-\frac{4\pi}{5}, \frac{4\pi}{5}]$. This extension allows us to control the decay rate of $m_0$ near $\frac{2\pi}{3}$, which yields global decay estimates for $\psi$ and $\hat\psi$. In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction.

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Filip Tomić, Stefan Tutić, Milica Žigić. 2025-12-05. Extending wavelet regularity beyond Gevrey classes. https://arxiv.org/abs/2512.05655

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