Nonstationary matrix-valued multiresolution analysis from the extended affine group

We characterize scaling functions of nonstationary matrix-valued multiresolution analysis in the matrix-valued function space \(L^2(\mathbb{R}, \mathbb{C}^{l \times l})\), l is a natural number. This is inspired by the work of Novikov, Protasov and Skopina on nonstationary multiresolution analysis of the space \(L^2(\mathbb{R})\). Using a sequence of diagonal matrix-valued scaling functions in \(L^2(\mathbb{R}, \mathbb{C}^{l \times l})\), the construction of matrixvalued nonstationary orthonormal wavelets associated with the affine group is presented. Nonstationary matrix-valued wavelet frames in terms of frames of closed subspaces associated with a given nonstationary multiresolution analysis are given. Finally, we give sufficient conditions for the sequence of scaling functions of nonstationary matrix-valued multiresolution analysis in the frequency domain.