Inequalities for Nonuniform Wavelet Frames

TL;DR

This paper constructs nonuniform wavelet frames based on spectral pairs, providing sufficient Fourier-based conditions for their existence, extending wavelet theory to non-discrete translation sets.

Contribution

It introduces new conditions for nonuniform wavelet frames linked to spectral pairs, broadening the scope of wavelet frame construction beyond traditional settings.

Findings

Established sufficient Fourier-based conditions for nonuniform wavelet frames.

Constructed wavelet frames associated with spectral pairs.

Extended wavelet theory to non-discrete translation sets.

Abstract

Gabardo and Nashed have studied nonuniform wavelets based on the theory of spectral pairs for which the associated translation set $\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z$ is no longer a discrete subgroup of $\mathbb R$ but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we construct the associated wavelet frames and establish some sufficient conditions that ensure the nonuniform wavelet system $\left\{ \psi_{j, \lambda}(x)= (2N)^{j/2}\psi\big( (2N)^{j}x-\lambda\big), j\in \mathbb Z, \lambda\in \Lambda \right\}$ to be a frame for $L^2(\mathbb R)$. The conditions proposed are stated in terms of the Fourier transforms of the wavelet system's generating functions.