On the existence and constructions of orthonormal wavelets on $L_2(\mathbb {R}^s)$

For a multiresolution analysis of $L _2 (\mathbb {R}^ s)$ associated with the scaling matrix $T$ having determinant $n$ we prove the existence of a wavelet basis with certain desirable properties if $2n-1 >s$ and its real-valued counterpart if the scaling function is real-valued and $n - 1 > s$. That those results cannot be extended to $2n - 1 \leq s$ and $n -1 \leq s$ respectively in general is demonstrated by Adams's theorem about vector fields on spheres. Moreover we present some new explicit constructions of wavelets, among which is a variation of Riemenschneider-Shen's method for $s\leq 3 .$