Unconditional convergence constants of Hilbert space frame expansions

We will prove some new, fundamental results in frame theory by computing the unconditional constant (for all definitions of unconditional) for the frame expansion of a vector in a Hilbert space and see that it is √B/A, where A, B are the frame bounds of the frame. It follows that tight frames have unconditional constant one. We then generalize this to a classification of such frames by showing that for Bessel sequences whose frame operator can be diagonalized, the frame expansions have unconditional constant one if and only if the Bessel sequence is an orthogonal sum of tight frames. We then prove similar results for cross frame expansions but here the results are no longer a classification. We also give examples to show that our results are best possible. These results should have been done 20 years ago but somehow we overlooked this topic.