A Construction of Multiframe Wavelet Sets in ${\mathbb{R}}^{2}$

Abstract

For an expansive matrix A, which preserves the lattice ${\mathbb{Z}}^{n}$ i.e. ${\text{A}}{\mathbb{Z}}^{n}\subset {\mathbb{Z}}^{n}$, we provide characterization for multiframe wavelet set in ${\mathbb{R}}^{n}$. We also obtain necessary condition for a set $\mathrm{G }\subset {\mathbb{R}}^{n}$ to be frame scaling set for expansive matrix A. By taking frame scaling set ${\text{S}}$ in ${\mathbb{R}}$ and frame wavelet set E in ${\mathbb{R}}$, we have constructed frame wavelet set of the form $\mathrm{E }\times {\text{S}}$ in ${\mathbb{R}}^{2}$, for a matrix of the form $\left(\begin{array}{cc}0&\quad 1\\ a&\quad 0\end{array}\right)$, where |a| > 1.