Excess of Parseval Frames

The excess of a sequence in a Hilbert space H is the greatest number of elements that can be removed yet leave a set with the same closed span. This paper proves that if F is a frame for H and there exist infinitely many elements gn ∈ F such that F \ {gn} is complete for each individual n and if there is a uniform lower frame bound L for each frame F \ {gn}, then for each ε > 0 there exists an infinite subsequence {gnk}k∈N of {gn}n∈N such that F \ {gnk}k∈N is still a frame for H . Moreover, if the frame is Parseval (i.e., has frame bounds A = B = 1), then we show that for each ε > 0 this can be done in a way that changes the lower frame bound to no less than L − ε.