Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Band-limited wavelets beyond Gevrey regularity

Nenad Teofanov, Filip Tomić, Stefan Tutić

Abstract

It is known that a smooth function of exponential decay at infinity can not be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey type subexponential decay. We weaken the Gevrey type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that prescise asymptotics of functions from such classes can be given in terms of the Lambert $W$ function.

Band-limited wavelets beyond Gevrey regularity

Abstract

It is known that a smooth function of exponential decay at infinity can not be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey type subexponential decay. We weaken the Gevrey type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that prescise asymptotics of functions from such classes can be given in terms of the Lambert function.
Paper Structure (8 sections, 8 theorems, 65 equations)

This paper contains 8 sections, 8 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The Lambert function
  5. Defining sequences
  6. Associated functions
  7. Extended Gevrey classes
  8. Main results

Key Result

Proposition 2.2

Let $\tau>0$, $\sigma>1$, $M^{\tau,\sigma}_p=p^{\tau p^\sigma}$, $p = 1,2, \dots$, let $T_{\tau,\sigma}$ be the associated function to the sequence ${M_p^{\tau,\sigma}}$, and put Then the following estimates hold: for large enough $k>0$, and suitable constants $A_{\sigma}, B_{\sigma}, \widetilde{A}_{\tau,\sigma},\widetilde{B}_{\tau,\sigma}>0$.

Theorems & Definitions (16)

  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4: PTT-01
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 6 more