Extending wavelet regularity beyond Gevrey classes

TL;DR

This paper constructs a smooth wavelet with Fourier transform properties extending beyond classical Gevrey classes, using invariant cycles and Lambert W functions to control decay rates.

Contribution

It introduces a wavelet whose Fourier transform and itself belong to an extended Gevrey class beyond all classical classes, using a novel extension of the low-pass filter support.

Findings

Wavelet and Fourier transform in extended Gevrey class $ ext{E}_\sigma$ for $\sigma > 1$

Decay estimates involve Lambert W function

Supports extended from $[-2 ext{ extperiodcentered}3, 2 ext{ extperiodcentered}3]$ to $[-4 ext{ extperiodcentered}5, 4 ext{ extperiodcentered}5]$

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.