On vector-valued nonuniform multiresolution analysis

In a vector-valued nonuniform multiresolution analysis (VNUMRA, in short), the corresponding translation set may not be a subgroup of 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.