Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Meyer wavelets for rational dilations

Marcin Bownik

TL;DR

This work advances multidimensional wavelet theory by resolving the existence and localization questions for MRAs and Meyer wavelets under general dilation structures. It first proves the existence of smooth, band-limited MRAs for any expansive real matrix $A$ in any dimension and then constructs orthonormal Meyer wavelets for rational dilations using matrix-filter bank techniques and polyphase decompositions. A sharp optimality result shows that well-localized MRA-based wavelets can exist only when the dilation has rational entries, underscoring fundamental limits in irrational settings. The paper also develops strictly expansive and minimal-generator frameworks, along with lifting methods to extend 1D rational constructions to higher dimensions, thereby broadening the applicability of Meyer-type bases to non-integer, high-dimensional settings.

Abstract

We show the existence of smooth band-limited multiresolution analysis (MRA) for any expansive dilation with real entries in any spatial dimension. We then prove the existence of orthonormal Meyer wavelets, which have smooth and compactly supported Fourier transform, for any expansive dilation with rational entries and any spatial dimension. This extends one dimensional results of Auscher. In a converse direction, we show that well-localized orthogonal MRA wavelets, such as Meyer wavelets, can only exist for expansive dilations with rational entries. This shows the optimality of our existence result and extends one dimensional result of Lemarié-Rieusset.

Meyer wavelets for rational dilations

TL;DR

This work advances multidimensional wavelet theory by resolving the existence and localization questions for MRAs and Meyer wavelets under general dilation structures. It first proves the existence of smooth, band-limited MRAs for any expansive real matrix in any dimension and then constructs orthonormal Meyer wavelets for rational dilations using matrix-filter bank techniques and polyphase decompositions. A sharp optimality result shows that well-localized MRA-based wavelets can exist only when the dilation has rational entries, underscoring fundamental limits in irrational settings. The paper also develops strictly expansive and minimal-generator frameworks, along with lifting methods to extend 1D rational constructions to higher dimensions, thereby broadening the applicability of Meyer-type bases to non-integer, high-dimensional settings.

Abstract

We show the existence of smooth band-limited multiresolution analysis (MRA) for any expansive dilation with real entries in any spatial dimension. We then prove the existence of orthonormal Meyer wavelets, which have smooth and compactly supported Fourier transform, for any expansive dilation with rational entries and any spatial dimension. This extends one dimensional results of Auscher. In a converse direction, we show that well-localized orthogonal MRA wavelets, such as Meyer wavelets, can only exist for expansive dilations with rational entries. This shows the optimality of our existence result and extends one dimensional result of Lemarié-Rieusset.

Paper Structure

This paper contains 7 sections, 33 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Symmetric frequency supported shift invariant basis in $L^2({\mathbb R^n})$
  3. Schwartz class MRA associated with real dilations
  4. Constructing wavelets from rationally dilated MRAs
  5. Matrix completion and existence of Meyer wavelets
  6. Strictly expansive dilations
  7. Limitations on the existence of well-localized wavelets

Key Result

Theorem 1.1

For every expansive $2\times 2$ integer dilation $A$, there exists a wavelet consisting of $(|\det A|-1)$ band-limited Schwartz class functions.

Figures (2)

  • Figure 1: Graphs of functions $\widehat{f}_j$, $j=0,1,2,3$.
  • Figure 2: Ellipsoid $\mathcal{E}$ and its image under expansive dilation $B$ superimposed with the supports of the SFS basis generators $\widehat{w}_{(j_1,j_2)}$ constituting the scaling vector $\Phi$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • ...and 59 more