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New Tight Wavelet Frame Constructions Sharing Responsibility

Youngmi Hur, Hyojae Lim

TL;DR

The paper addresses the challenge of constructing tight wavelet frames in $L^2(\mathbb{R}^n)$ when SOS-based methods (sub-QMF and oblique sub-QMF) require explicit SOS generators for wavelet masks. It introduces a novel framework that distributes the construction burden between refinable masks and wavelet masks, enabling explicit, SOS-friendly TWFs without solving the SOS problem in isolation. Two main branches are developed: (i) a sub-QMF construction with a trig polynomial refinement mask and (ii) an oblique sub-QMF construction with a rational trig polynomial refinement mask, each with explicit formulas for the MRA masks and vanishing-moment analysis. The paper also proves well-definedness and $L^2$-convergence of refinable functions arising from rational refinement masks and provides multiple examples to illustrate the method’s flexibility and potential impact on practical frame constructions.

Abstract

Tight wavelet frames (TWFs) in \(L^2(\mathbb{R}^n)\) are versatile, and are practically useful due to their perfect reconstruction property. Nevertheless, existing TWF construction methods exhibit limitations, including a lack of specific methods for generating mother wavelets in extension-based construction, and the necessity to address the sum of squares (SOS) problem even when specific methods for generating mother wavelets are provided in SOS-based construction. Many TWF constructions begin with a given refinable function. However, this approach places the entire burden on finding suitable mother wavelets. In this paper, we introduce TWF construction methods that spread the burden between both types of functions: refinable functions and mother wavelets. These construction methods offer an alternative approach to addressing the SOS problem. We present examples to illustrate our construction methods.

New Tight Wavelet Frame Constructions Sharing Responsibility

TL;DR

The paper addresses the challenge of constructing tight wavelet frames in when SOS-based methods (sub-QMF and oblique sub-QMF) require explicit SOS generators for wavelet masks. It introduces a novel framework that distributes the construction burden between refinable masks and wavelet masks, enabling explicit, SOS-friendly TWFs without solving the SOS problem in isolation. Two main branches are developed: (i) a sub-QMF construction with a trig polynomial refinement mask and (ii) an oblique sub-QMF construction with a rational trig polynomial refinement mask, each with explicit formulas for the MRA masks and vanishing-moment analysis. The paper also proves well-definedness and -convergence of refinable functions arising from rational refinement masks and provides multiple examples to illustrate the method’s flexibility and potential impact on practical frame constructions.

Abstract

Tight wavelet frames (TWFs) in \(L^2(\mathbb{R}^n)\) are versatile, and are practically useful due to their perfect reconstruction property. Nevertheless, existing TWF construction methods exhibit limitations, including a lack of specific methods for generating mother wavelets in extension-based construction, and the necessity to address the sum of squares (SOS) problem even when specific methods for generating mother wavelets are provided in SOS-based construction. Many TWF constructions begin with a given refinable function. However, this approach places the entire burden on finding suitable mother wavelets. In this paper, we introduce TWF construction methods that spread the burden between both types of functions: refinable functions and mother wavelets. These construction methods offer an alternative approach to addressing the SOS problem. We present examples to illustrate our construction methods.
Paper Structure (11 sections, 10 theorems, 82 equations)

This paper contains 11 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tight wavelet frames and the extension principles
  4. Related works
  5. Construction of tight wavelet frames sharing responsibility
  6. New tight wavelet frames with trigonometric polynomial refinement masks
  7. Examples
  8. New tight wavelet frames with rational trigonometric polynomial refinement masks
  9. On refinable functions with rational trigonometric polynomial refinement masks
  10. Conclusion
  11. Proofs of auxiliary results in Section \ref{['subsubsection: examples of our sub-QMF method']}

Key Result

Theorem 3.1

Let $N \leq \lambda^n$. For each $1 \leq l \leq N$, suppose that there exist $r_l \in \mathbb{N}$ and trigonometric polynomials $p_l$ and $g_l^{(r)}$, $1 \leq r \leq r_l$, satisfying with $p_l(0)=1$. In case of $N<\lambda^n$, we set $p_l$ a trigonometric polynomial satisfying $\lvert p_l \rvert^2 = 1$ for $l=N+1, \cdots, \lambda^n$. We define a trigonometric polynomial $\tau$ as where $\nu_1, \c

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Remark 2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • ...and 21 more