Inequalities for Nonuniform Wavelet Frames

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.