Redundancy for localized frames (Q1758942)

Bodmann, Casazza and Kutyniok have introduced a \textit{quantitative} notion of redundancy for finite frames. The same for the case of \textit{infinite} frames has proven to be elusive. In this article the authors claim to have achieved this for the infinite frame setting. Using the \textit{density} of a frame as a measure for redundancy, the main result is the following. For any \(0 < \epsilon < 1,\) every \(\ell^1\)-localized frame with frame density \(d > 1\) has a subframe with frame density smaller than \(1 + \epsilon.\) This together with other work establishes frame density as a quantitative definition of redundancy. The above result connects redundancy of a frame to the ability to remove large number of elements from the frame and still have the remaining elements form a frame. The result is then specialized to Gabor frames with generalizations to Gabor multi-frames and Gabor molecules. The work is based on reducing the infinite dimensional case to a sequence of finite-dimensional cases and using a fundamental result for the finite-dimensional setting.