Existence of multiwavelets in $\mathbb{R} ^n$

For a $q$-regular Multiresolution Analysis of multiplicity $r$ with arbitrary dilation matrix $A$ for a general lattice $\Gamma$ in $\mathbb{R} ^n$, we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if $2r(m-1) \geq n$ where $m$ is the absolute value of the determinant of $A$, then these conditions are always met, and therefore an associated wavelet basis of $q$-regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix $A$ for a lattice $\Gamma$.