Nonstationary matrix-valued multiresolution analysis from the extended affine group

We characterize scaling functions of nonstationary matrix-valuedmultiresolution analysis in the matrix-valued function space \(L^2(\mathbb{R}, \mathbb{C}^{l \times l})\) $L^2(\mathbb{R},{\mathbb{C}}^{l\times l})$ , l is a naturalnumber. This is inspired by the work of Novikov, Protasov and Skopina onnonstationary multiresolution analysis of the space \(L^2(\mathbb{R})\) $L^2(\mathbb{R})$ . Using a sequence of diagonalmatrix-valued scaling functions in \(L^2(\mathbb{R}, \mathbb{C}^{l \times l})\) $L^2(\mathbb{R},{\mathbb{C}}^{l\times l})$ , the construction of matrixvaluednonstationary orthonormal wavelets associated with the affine group ispresented. Nonstationary matrix-valued wavelet frames in terms of frames ofclosed subspaces associated with a given nonstationary multiresolution analysisare given. Finally, we give sufficient conditions for the sequence of scaling functionsof nonstationary matrix-valued multiresolution analysis in the frequencydomain.