Local theory of frames and Schauder bases for Hilbert space

Abstract

We develope a local theory for frames on finite-dimensional Hilbert spaces. We show that for every frame (fi)mi=1 for an n-dimensional Hilbert space, and for every ∈ > 0, there is a subset I ⊂ {1, 2, . . . , m} with |I| ≥ (1 - ∈)n so that (fi)i∈I is a Riesz basis for its span with Riesz basis constant a function of ∈, the frame bounds, and (||fi||)mi=1, but independent of m and n. We also construct an example of a normalized frame for a Hilbert space H which contains a subset which forms a Schauder basis for H, but contains no subset which is a Riesz basis for H. We give examples to show that all of our results are best possible, and that all parameters are necessary.