Standard pairs and construction of multiwavelets using refinement masks satisfying sum rules of order one

Abstract

A multiwavelet is typically constructed starting from a vector-valued function satisfying a matrix refinement equation. The approximation order of such a refinable function vector is related to the sum rules of order [Formula: see text] satisfied by the corresponding refinement mask. A refinable function vector can be obtained using cascade algorithm by constructing a refinement mask which satisfies the sum rules of order [Formula: see text] A standard pair associated with a refinement mask gives information about its spectral properties. In this paper, we present a procedure for constructing refinement masks satisfying the sum rules of order [Formula: see text], starting from standard pairs. How this helps in the construction of asymmetric multiwavelets using standard pairs is illustrated through examples. A sufficient condition on a standard pair and a necessary and sufficient condition on a left standard pair are established so that the corresponding refinement mask satisfies the sum rules of order [Formula: see text].