A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of \({\mathbb {R}}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we continue the study based on this nonstandard setting and introduce fractional vector-valued nonuniform multiresolution analysis (Fr-VNUMRA) where the associated subspace of the function space has an orthonormal basis. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of fractional vector-valued nonuniform multiresolution analysis starting from a vector refinement mask with appropriate conditions. Nevertheless, to extend the scope of the present study, we worked to construct the associated wavelet packets for such an MRA and investigate their properties by means of fractional Fourier transform.

在 Gabardo 和 Nashed 的两篇文章中，考虑了 Mallat 经典多分辨率分析的推广，该分析基于光谱对理论。在此设置中，关联的翻译集不再是\({\mathbb {R}}\)的离散子群但是与某个一维谱对相关联的谱和相关的膨胀是与给定谱对相关的偶正整数。在本文中，我们继续基于这种非标准设置进行研究，并引入分数向量值非均匀多分辨率分析（Fr-VNUMRA），其中函数空间的相关子空间具有正交基。我们建立了关联小波存在的充分必要条件，并推导出了一种从具有适当条件的向量细化掩模开始构建分数向量值非均匀多分辨率分析的算法。然而，为了扩大本研究的范围，