On matrix-valued Gabor Bessel sequences and dual frames over locally compact abelian groups

Document Type : Original Article


Department of Mathematics, Shivaji College, University of Delhi, Delhi-110027, India



We study matrix-valued Gabor Bessel sequences and frames in the matrix-valued space $L^2(G, \mathbb{C}^{n\times n})$, where $G$ is a locally compact abelian group and $n$ is a positive integer. Firstly, we show that the Bessel condition (or upper frame condition) can be extended from $L^2(G)$ to its associated matrix-valued signal space $L^2(G,\mathbb{C}^{n\times n})$, and conversely. However, this is not true for the lower frame condition. Secondly, we give sufficient conditions for the extension of a pair of matrix-valued Bessel sequences to matrix-valued dual frames over LCA groups. A special class of matrix-valued dual generators is given. It is shown that the symmetric windows associated with a given matrix-valued Gabor frames constitutes a Gabor frame in matrix-valued spaces over LCA groups.