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 ${[0,1]}^d$ 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 $[0,1]$ 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 $[0,\infty )$ 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. 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 $[0,1]$ will be provided to illustrate our construction approaches and proposed algorithms.

实线上的正交和双正交（多）子波已经得到了广泛的研究，并在成功的应用中得到了应用。另一方面，在有界区间或域上定义了许多应用程序中的问题，例如图像和微分方程的解。因此，在理论上和应用上都重要的是，以实线上的（双）正交（多）小波为基础，以一定间隔构造所有可能的小波，并具有某些所需的特性。然后在矩形域上的小波${[0，\mathrm{1个}]}^d$可以通过张量积获得。紧支撑小波的消失矩是稀疏小波表示的关键属性，并且与其基础可精化（矢量）函数的多项式再现紧密相关。具有低阶消失矩的边界小波通常会导致不期望的边界伪像，并导致应用中边界附近的稀疏性和近似阶数减少。区间上的标量正交小波和样条双正交小波$[0，\mathrm{1个}]$在文献中已经进行了广泛的研究。尽管多小波在标量小波上具有某些期望的属性，例如高消失矩和相对短的支持，但除了一些具体示例外，目前尚无系统的方法来构造有界区间上的（双）正交多小波。与当前关于从特殊（双）正交（多）小波以一定间隔构造特定小波的文献，在实线上任意给定的紧支持（双）正交多小波的文献相反，在本文中，我们提出了两种不同的构造/推导方法上所有可能的本地支持的（双）正交（多）小波$[0，\infty ）$ 或者 $[0，\mathrm{1个}]$有或没有规定的消失力矩，多项式重现和/或齐次边界条件。第一种方法将经典方法从标量小波推广到多小波，而第二种方法是直接方法，没有明确涉及任何双重可精函数和双重多小波。我们还将在满足一般齐次边界条件的区间上处理小波。尽管在区间上构造正交（多）子波要比双正交小波容易得多，但我们证明，如果这些区间上的正交（多）子波满足齐次Dirichlet边界条件，那么某些边界正交子波就不会有消失的矩。与经典方法相比，我们提出的直接方法使在间隔上轻松构建所有可能的局部支持（多）小波成为可能。区间上正交和双正交多小波的七个示例$[0，\mathrm{1个}]$ 将提供以说明我们的构造方法和建议的算法。