Perturbation of frame sequences in shift-invariant spaces (Q2577085)

Given two subspaces \(V\) and \(W\) of a Hilbert space with \(V \neq \{0\}\), the \textit{gap} from \(V\) to \(W\) is defined by \[ \delta(V,W):= \sup_{x \in V, \|x\| =1} \text{dist} (x, W) =\sup_{x \in V, \| x\|=1} \inf_{y \in W} \|x-y\| . \] In the last decade or so, perturbation questions for frames in Hilbert spaces have been studied extensively for general frames as well as for Gabor frames and wavelet frames. Some results have also been obtained for frame sequences, i.e., frames that only span a closed subspace of a given Hilbert space; however, all of these are based on a calculation of the gap between two subspaces, and are quite complicated to apply. The authors of the present article overcome this difficulty by using the so-called infimum cosine angle between the relevant subspaces. This allows them to improve a standard result on frame perturbation. Using Gramian analysis, they also obtain new perturbation results for shift--invariant subspaces.