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Wavelets on intervals derived from arbitrary compactly supported biorthogonal multiwavelets
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-03-03, DOI: 10.1016/j.acha.2021.02.006
Bin Han, Michelle Michelle

Orthogonal and biorthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. On the other hand, a lot of problems in applications such as images and solutions of differential equations 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. Then wavelets on rectangular domains such as ${\left[0,1\right]}^{d}$ can be obtained through tensor product. 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. Scalar orthogonal wavelets and spline biorthogonal wavelets on the interval $\left[0,1\right]$ have been extensively studied in the literature. Though multiwavelets enjoy some desired properties over scalar wavelets such as high vanishing moments and relatively short support, except a few concrete examples, there is currently no systematic method for constructing (bi)orthogonal multiwavelets on bounded intervals. In contrast to current literature on constructing particular wavelets on intervals from special (bi)orthogonal (multi)wavelets, 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 $\left[0,\infty \right)$ or $\left[0,1\right]$ 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. We shall also address wavelets on intervals satisfying general homogeneous boundary conditions. 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. In comparison with the classical approach, our proposed direct approach makes the construction of all possible locally supported (multi)wavelets on intervals easy. Seven examples of orthogonal and biorthogonal multiwavelets on the interval $\left[0,1\right]$ will be provided to illustrate our construction approaches and proposed algorithms.