Biorthogonal multiwavelets on the interval: Cubic Hermite splines

Starting with Hermite cubic splines as the primal multigenerator, first a d ual multigenerator on R is constructed that consists of continuous function s, has small support, and is exact of order 2. We then derive multiresoluti on sequences on the interval while retaining the polynomial exactness on th e primal and dual sides. This guarantees moment conditions of the correspon ding wavelets. The concept of stable completions [CDP] is then used to cons truct the corresponding primal and dual multiwavelets on the interval as fo llows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compac tly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algor ithms are simple and efficient. Furthermore, in addition to the Jackson est imates which follow from the erectness, one can also show Bernstein inequal ities for the primal and dual multiresolutions. Consequently, sequence norm s for the coefficients based on such multiwavelet expansions characterize S obolev norms parallel to . parallel to(Hs([0,1])) for s is an element of (- 0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L-2([ 0, 1]).