Wavelets on Intervals Derived from Arbitrary Compactly Supported Biorthogonal Multiwavelets

TL;DR

This paper introduces two methods to construct biorthogonal multiwavelets on bounded intervals from those on the real line, addressing boundary issues and vanishing moments for improved sparse representations and applications.

Contribution

It proposes two novel approaches for deriving all possible biorthogonal multiwavelets on intervals from given wavelets on the real line, including boundary condition considerations.

Findings

Constructed multiwavelets with prescribed properties on intervals.

Demonstrated that some orthogonal wavelets cannot have vanishing moments under boundary conditions.

Provided examples illustrating the construction methods.

Abstract

(Bi)orthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. A lot of problems in applications are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. From any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on $[0,\infty)$ or $[0,1]$ with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. (Multi)wavelets on intervals satisfying homogeneous boundary conditions will also be addressed. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. Several examples of orthogonal and biorthogonal multiwavelets on the interval $[0,1]$ will be provided to illustrate our construction approaches and proposed algorithms.