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

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