Characterization of scaling functions in a multiresolution analysis

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map $A: \mathbb{R} ^n\rightarrow \mathbb{R} ^n$ such that $A(\mathbb{Z} ^n) \subset \mathbb{Z} ^n$ and all (complex) eigenvalues of $A$ have absolute value greater than $1.$ In the general case the conditions depend on the map $A.$ We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when $A$is a diagonal matrix with all numbers in the diagonal equal to $2.$ There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.