Unitary Extension Principle for Nonuniform Wavelet Frames in $L^2(\mathbb{R})$

TL;DR

This paper develops methods to construct nonuniform tight wavelet frames in $L^2( eal)$ using the unitary extension principle, even when translation sets are not groups, with illustrative examples.

Contribution

It extends the UEP and OEP frameworks to nonuniform wavelet frames with non-group translation sets in $L^2( eal)$.

Findings

Established UEP and OEP for nonuniform wavelet frames

Constructed multi-generated tight wavelet frames in $L^2( eal)$

Provided examples illustrating the theoretical results

Abstract

We study the construction of nonuniform tight wavelet frames for the Lebesgue space $L^2(\mathbb{R})$, where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension principle (UEP) and the oblique extension principle (OEP) for construction of multi-generated nonuniform tight wavelet frames for $L^2(\mathbb{R})$. Some examples are also given to illustrate the results.