Characterization of wavelets associated with $AB$-MRA on $L^2(\mathbb R^n)$

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2(\mathbb R^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are matrices whose eigenvalues have magnitude one, while the members of $A$ are matrices expanding on a proper subspace of $\mathbb R^n$. In this paper, we provide the characterization of composite wavelets based on results of affine and quasi affine frames. Furthermore all the composite wavelets associated with $AB$-MRA on $L^2(\mathbb R^n)$ are also characterized.