Extending wavelet regularity beyond Gevrey classes

Abstract

We construct a smooth orthonormal wavelet $\ddot{I}ˆ$ such that both $\ddot{I}ˆ$ and its Fourier transform $\hat{\ddot{I}}ˆ$ belong to the extended Gevrey class ${\mathcal{E}}_{\ddot{I}}ƒ(\mathbb{R})$ for $\ddot{I}ƒ>1$, 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 $[-\frac{2\ddot{I}€}{3},\frac{2\ddot{I}€}{3}]$ to $[-\frac{4\ddot{I}€}{5},\frac{4\ddot{I}€}{5}]$. This extension allows us to control the decay rate of $m_0$ near $\frac{2\ddot{I}€}{3}$, which yields global decay estimates for $\ddot{I}ˆ$ and $\hat{\ddot{I}}ˆ$. In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction.