Approximately dual pairs of wavelet frames

TL;DR

This paper investigates the structural limitations of wavelet frames and their duals, demonstrating conditions under which approximate duals can be constructed to achieve near-perfect reconstruction, even when exact duals with wavelet structure do not exist.

Contribution

It proves the non-existence of wavelet-structured duals for certain frames and shows how mild decay conditions enable the construction of nearly perfect approximate duals with oversampling.

Findings

Existence of wavelet frames with no wavelet-structured duals.

Under decay conditions, oversampled frames admit approximate duals.

Large oversampling parameter N yields near-perfect reconstruction.

Abstract

This paper deals with structural issues concerning wavelet frames and their dual frames. It is known that there exist wavelet frames $\{a^{j/2}\psi( a^j\cdot -kb)\}_{j,k\in \mathbb Z}$ in $L^2(\mathbb R)$ for which no dual frame has wavelet structure. We first generalize this result by proving that there exist wavelet frames for which no approximately dual frame has wavelet structure. Motivated by this we show that by imposing a very mild decay condition on the Fourier transform of the generator $\psi\in L^2(\mathbb R),$ a certain oversampling $\{a^{j/2}\psi( a^j\cdot -kb/N)\}_{j,k\in \mathbb Z}$ indeed has an approximately dual wavelet frame; most importantly, by choosing the parameter $N\in \mathbb N$ sufficiently large we can get as close to perfect reconstruction as desired, which makes the approximate dual frame pairs perform equally well as the classical dual frame pairs in applications.