Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$

TL;DR

This paper presents a method to construct dual wavelet frames in $L^2(R^n)$ using bandlimited functions with specific Fourier properties, enabling explicit dual pairs with compact Fourier support.

Contribution

It introduces a new construction technique for dual wavelet frames in higher dimensions based on bandlimited functions satisfying a partition of unity condition.

Findings

Explicit dual wavelet frames can be constructed with compact Fourier support.

The construction provides a simple procedure for dual pairs with desired time localization.

A general class of functions satisfying the technical condition is identified.

Abstract

Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame generated by a finite linear combination of dilations of $\psi$ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction relies on a technical condition on $\psi$, and we exhibit a general class of function satisfying this condition.