On matrix-valued wave packet frames in $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$

Abstract

In this paper, we study matrix-valued wave packet frames for the matrix-valued function space $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$ . An interplay between matrix-valued wave packet frames and its associated atomic wave packet frames is discussed. This is inspired by examples which show that frame properties cannot be carried from matrix-valued wave packet scaling functions to its associated atomic wave packet scaling functions and vice versa. Construction of matrix-valued wave packet frames for $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$ from corresponding atomic wave packet frames for $$L^2({\mathbb {R}}^d)$$ (and conversely) are given. Some special classes of matrix-valued scaling functions are given. A characterization of tight matrix-valued wave packet frames in terms of orthogonality of Bessel sequences has been obtained. Further, we provide a characterization of superframes which can generate matrix-valued frames. Finally, a Paley-Wiener type perturbation result with respect to matrix-valued wave packet scaling functions is given.