Intertwining Multiresolution Analyses and the Construction of Piecewise-Polynomial Wavelets

Let $(V_p )$ be a local multiresolution analysis of $L^2 ({\bf R})$ of multiplicity $r \geq 1$, i.e., $V_0 $ is generated by $r$ compactly supported scaling functions. If the scaling functions generate an orthogonal basis of $V_0 $, then $(V_p )$ is called an orthogonal multiresolution analysis. We prove that there exists an orthogonal local multiresolution analysis $(V'_p )$ of multiplicity $r'$ such that \[V_q \subset V'_0 \subset V_{q + n} \] for some integers $q \geq 0$, $n \geq 1$, and $r' > 1$. In particular, this shows that compactly supported orthogonal polynomial spline wavelets and scaling functions (of multiplicity $r' > 1$) of arbitrary regularity exist, and we give several such examples.