Banded frames with banded duals

Abstract

Banded invertible matrices typically do not have banded inverses, but the case when the inverse is banded is characterized by a factorization into block diagonal matrices. In this paper, we extend this result to full rank but non-invertible banded matrices. Such matrices have either a left or right inverse, but not both. These matrices arise naturally in frame theory, where a surjective matrix corresponds to a frame, and its right inverses correspond to dual frames. We generalize a theorem of Asplund and apply it to describe banded matrices with banded left or right inverses. Equivalently, we characterize banded finite frames with banded dual frames.