Interpolatory orthogonal multiwavelets and refinable functions

Abstract

Multiwavelet bases of L/sub 2/ consist of families of functions {2/sup j/2//spl psi//sub 2/(2/sup j/x-k)}. By allowing more than one function {/spl psi//sub 1/,/spl psi//sub 2/}, multiwavelets provide some useful applications in signal processing and nice features such as symmetry and orthogonality. The elementary structure for multiwavelets is the multiresolution analysis of multiplicity two {V/sub j/} generated by dilating the basic subspace V/sub 0/. This subspace V/sub 0/ is generated by a multiple refinable function /spl phi/=(/spl phi//sub 1/,/spl phi//sub 2/)/sup T/ (refinable vector of functions) satisfying a vector refinement equation /spl phi/(x)=/spl Sigma/a(k)/spl phi/(2x-k). Here, each a(k) is a 2/spl times/2 matrix. In this paper, we investigate interpolatory orthogonal multiple refinable functions and multiwavelets. The interpolatory property here means that /spl phi//sub 1/ and /spl phi//sub 2/ vanish at all integers and half integers, except that /spl phi//sub 1/(0)=1 and /spl phi//sub 2/(1/2)=1. When /spl phi/ is both interpolatory and orthogonal (which is impossible for scalar refinable functions), the coefficients in the multiresolution representation can be realized by sampling instead of inner products. If f(x)=/spl Sigma/{c/sub 1/(k)/spl phi//sub 1/(2/sup N/x-k)+c/sub 2/(k)/spl phi//sub 2/(2/sup N/x-k)}, then c/sub 1/(k)=f(k/2/sup N/) and c/sub 2/(k)=f(k/2/sup N/+1/2/sup N+1/) for k/spl isin/Z. What is more, the orthogonal multiwavelets we construct here are also interpolatory. We show that the refinement mask for an interpolatory orthogonal multiple refinable function and multiwavelets (filterbank) is reduced to a scalar CQF. The approximation order of interpolatory multiple refinable functions is described. A complete characterization of interpolatory orthogonal multiple refinable functions is given in this paper. However, interpolatory orthogonal multiple refinable functions cannot be symmetric. Examples are presented to illustrate the general theory.