POLLEN PRODUCT FACTORIZATION AND CONSTRUCTION OF HIGHER MULTIPLICITY WAVELETS

For the design of regular higher multiplicity wavelets it is useful to specify matrices of wavelet coefficients by their first row. This still leaves some freedom in the construction. In the case of classical wavelets (i.e., the wavelet matrix has only two rows), it means that a suitable characteristic matrix (the sum of square blocks) can be chosen. It is shown, however, that for m > 2 rows, given such data, the uniqueness fails, and when m greater than or equal to 4 there are infinitely many possibilities. They can all be described by choices of some nontrivial linear subspaces in an m-dimensional space. This leads to a simple, explicit, and numerically reliable algorithm for constructing any of them. On the way, the existence and uniqueness of the factorisation of wavelet matrices with respect to the Pollen product is also resolved.