Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets (Q5937253)

In this interesting paper, the author investigates the approximation properties of a multivariate refinable function vector. Based on the sum rules of biorthogonal multiwavelets, a coset by coset (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. Given any primal matrix mask \(a\) and a dilation matrix \(M\), the proposed CBC algorithm reduces the construction of all dual masks of \(a\), which satisfy the sum rules of arbitrary order, to a problem of solving a system of linear equations. It is proved that for any given primal mask \(a\) with a dilation matrix \(M\) and for any positive integer \(k\), one can always construct a dual mask \(\widetilde a\) such that \(\widetilde a\) satisfies the sum rules of order \(k\). Further, a family of Hermite interpolatory masks is constructed. As an example, a \(C^3\) Hermite interpolant with support \([-3,3]\) is presented. Finally, several examples of biorthogonal multiwavelets are presented. In particular, a \(C^1\) dual function vector with support \([-4,4]\) of the cubic Hermite splines is given.