Balanced multi-wavelets in $\mathbb R^s$

The notion of $K$-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order $K$ (or degree $K-1$), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for $K=1$ is known. In addition, the formulation of the $K$-balancing condition for $K\ge 2$ is so prohibitively difficult to satisfy that only a very few examples for $K=2$ and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the $K$-balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any $K\ge 1$.