A construction of multiwavelets

Abstract

A class of r -regular multiwavelets, depending on the smoothness of the multiwavelet functions, is introduced with the appropriate notation and definitions. Oscillation properties of orthonormal systems are obtained in Lemma 1 and Corollary 1 without assuming any vanishing moments for the scaling functions, and in Theorem 1 the existence of r -regular multiwavelets in L 2 ( R n ) is established. In Theorem 2, a particular r -regular multiresolution analysis for multiwavelets is obtained from an r -regular multiresolution analysis for uniwavelets. In Theorem 3, an r -regular multiresolution analysis of split-type multiwavelets, which are perhaps the simplest multiwavelets, is easily obtained by using an r -regular multiresolution analysis for uniwavelets and a (2 n − 1)-fold regular multiresolution analysis for uniwavelets. For some split-type multiwavelets, the support or width of the wavelets is shorter than the support or width of the scaling functions without loss of regularity nor of vanishing moments. Examples of split-type multiwavelets in L 2 ( R ) are constructed and illustrated by means of figures. Symmetry and antisymmetry are preserved in the case of infinite support.