Reordered Frames and Weavings

Abstract

For two frames $\{\phi _i\}_{i \in {\mathcal {I}}}$ and $\{\psi _i\}_{i\in {\mathcal {I}}}$, the family $\{\phi _i\}_{i \in \sigma } \cup \{\psi _i\}_{i \in \sigma ^c}$ is called a weaving, where ${\mathcal {I}}$ is a countable index set and $\sigma \subset {\mathcal {I}}$. Two frames $\{\phi _i\}_{i \in {\mathcal {I}}}$ and $\{\psi _i\}_{i\in {\mathcal {I}}}$ are called woven if all their weavings are frames with the same frame bounds. The concept of this manuscript is motivated by an important and interesting problem which is, under what conditions the frames $\{\phi _i\}_{i \in {\mathcal {I}}}$ and $\{\phi _{\pi (i)}\}_{i\in {\mathcal {I}}}$ are woven for the separable Hilbert space H, where $\pi$ is a permutation function on ${\mathcal {I}}$. In this paper, the effect of reordering of the elements of weavings for frames is studied. In particular, full spark frames and m-uniform excess frames are studied. Also, an interesting characterization of reordered weavings by orthogonal projections is given. Finally, the effects of perturbations are considered.