From Wavelets to Multiwavelets

Abstract

. This paper gives an overview of recent achievements of the multiwavelet theory. The construction of multiwavelets is based on a multiresolution analysis with higher multiplicity generated by a scaling vector. The basic properties of scaling vectors such as L 2 -stability, approximation order and regularity are studied. Most of the proofs are sketched. 1. Introduction Wavelet theory is based on the idea of multiresolution analysis (MRA). Usually it is assumed that an MRA is generated by one scaling function, and dilates and translates of only one wavelet # # L 2 (IR) form a stable basis of L 2 (IR). This paper considers a recent generalization allowing several wavelet functions # 1 , . . . , # r . The vector ## # # # # # ## = (# 1 , . . . , # r ) T is then called a multiwavelet. Multiwavelets have more freedom in their construction and thus can combine more useful properties than the scalar wavelets. Symmetric scaling functions constructed by Geronimo, Hardin, and Massop..