Construction of Multiwavelets on an Interval

Abstract

Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions

Gauss-type quadrature rule with complex nodes and weights for integrals involving Daubechies scale functions and wavelets

“…Theoretical aspects of determination of boundary filters were considered in detail by Cohen et al [15], Andersson et al [16], Chui and Quak [17]. Their computational aspects have been discussed by Chyzak et al [18], Lee and Kassim [19] and Altürk and Keinert [20,21]. Now the vectors formed by the moments of boundary scale functions and wavelets are defined by…”

Boundary functions for wavelets and their properties

“…In fact, the necessity of choosing suitable matrices is what motivated the work in chapter 3. The results obtained in this chapter will appear in [2], [3].…”

“…We also demonstrate by an example that this is not the case when we have a boundary function vector -more than one boundary function (see chapter 5). The work contained in this chapter will appear in [2], [3].…”

“…The cascade algorithm converges in a single step and determines ϕ L (x) on [1,2]. The continuity of ϕ carries over to ϕ L .…”