Extending wavelet regularity beyond Gevrey classes

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.

💡 Research Summary

The paper presents a constructive method for obtaining a smooth orthonormal wavelet ψ whose both time‑domain function and Fourier transform ˆψ belong to the extended Gevrey class Eσ(ℝ) for any σ > 1, thereby providing an explicit example that lies beyond all classical Gevrey classes. The authors start from the classical multiresolution analysis (MRA) framework: a scaling function φ satisfies φ̂(ξ)=∏_{j≥1} m₀(2^{-j}ξ) and the associated wavelet is given by ˆψ(ξ)=e^{iξ/2} m₀(ξ/2+π) φ̂(ξ/2). Consequently, the regularity of ψ and ˆψ is entirely dictated by the low‑pass filter m₀.

The novelty lies in the manipulation of invariant cycles of the doubling map ρ(ξ)=2ξ (mod 2π). Earlier constructions (e.g., Lemarié‑Meyer) forced m₀ to vanish on a whole neighbourhood of the invariant points ±2π/3, producing band‑limited (hence analytic) wavelets. In contrast, the present work keeps the zeros at ±2π/3 but only requires a controlled, non‑exponential decay of m₀ in a small neighbourhood of these points. To achieve this, the support of m₀ is extended from the classical interval