Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending wavelet regularity beyond Gevrey classes

Filip Tomić, Stefan Tutić, Milica Žigić

TL;DR

This work addresses the existence of smooth, orthonormal wavelets with regularity beyond all classical Gevrey classes. By leveraging invariant cycles to extend the low-pass filter support and employing decay controls governed by functions built with the Lambert $W$ function, the authors construct a wavelet whose time-domain and frequency-domain profiles lie in the extended Gevrey class $\mathcal{E}_{\sigma}(\mathbb{R})$ with $\sigma>1$. They prove a sharp regularity result: $\psi$ and $\widehat{\psi}$ belong to $\mathcal{E}_{\sigma}$ but not to any finer $\mathcal{E}_{\sigma'}$, while the same holds for the Fourier side in a coordinated way. This demonstrates the feasibility of wavelets with intermediate regularity between Gevrey and $C^{\infty}$, offering new insights into ultradifferentiable function theory and wavelet design through Lambert $W$-based decay control.

Abstract

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.

Extending wavelet regularity beyond Gevrey classes

TL;DR

This work addresses the existence of smooth, orthonormal wavelets with regularity beyond all classical Gevrey classes. By leveraging invariant cycles to extend the low-pass filter support and employing decay controls governed by functions built with the Lambert function, the authors construct a wavelet whose time-domain and frequency-domain profiles lie in the extended Gevrey class with . They prove a sharp regularity result: and belong to but not to any finer , while the same holds for the Fourier side in a coordinated way. This demonstrates the feasibility of wavelets with intermediate regularity between Gevrey and , offering new insights into ultradifferentiable function theory and wavelet design through Lambert -based decay control.

Abstract

We construct a smooth orthonormal wavelet such that both and its Fourier transform belong to the extended Gevrey class for , 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 from to . This extension allows us to control the decay rate of near , which yields global decay estimates for and . In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction.

Paper Structure

This paper contains 8 sections, 15 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularity classes
  4. An example of the extended Gevrey function
  5. Wavelet construction
  6. Illustrations
  7. Appendix
  8. Acknowledgments

Key Result

Theorem 1.2

Let $\sigma>1$. For $\eta>1$ set $\sigma_{\eta}=\frac{\sigma+\eta (\sigma -1)}{1+\eta (\sigma-1)}\in(1,\sigma_0)$ where $\sigma_0=\max\{2,\sigma\}$. Then there exists an orthonormal wavelet $\psi$ such that

Figures (5)

  • Figure 1: The plot of the low-pass filter $m_0$ given in \ref{['NasLowPass']}. Note that it features a small bump near $\tfrac{2\pi}{3}$, which is a consequence of controlling its local decay.
  • Figure 2: Plot of the Fourier transform of the scaling function; $\hat{\varphi}$.
  • Figure 3: Note that $\hat{\varphi}$ inherits the bumps of $m_0$ near the invariant cycle points. Therefore $\hat{\varphi}$ is not band-limited.
  • Figure 4: Plot of Fourier transform of the wavelet; $\hat{\psi}$
  • Figure 5: Plot of the wavelet $\psi$

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 29 more