  (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.
