In a vector-valued nonuniform multiresolution analysis (VNUMRA, in short), the corresponding translation set may not be a subgroup of $\mathbb{R}$, but a spectrum associated with one-dimensional spectral pair. In this work, first, we give necessary and sufficient conditions for a family of functions related to VNUMRA to be an orthonormal system and complete system. We provide algebraic results related to the construction of associated vector-valued wavelets from the given VNUMRA. Finally, we give a sufficient condition for the construction of vector-valued wavelets from a given VNUMRA.

中文翻译：

---

关于向量值非均匀多分辨率分析

在向量值非均匀多分辨率分析（简称 VNUMRA）中，对应的平移集可能不是$\mathbb{R}$，而是与一维光谱对相关联的光谱。在这项工作中，我们首先给出了与VNUMRA相关的函数族成为正交系统和完备系统的充要条件。我们提供了与从给定 VNUMRA 构建相关向量值小波相关的代数结果。最后，我们给出了从给定 VNUMRA 构造向量值小波的充分条件。