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. $(f_{i})^{m}_{i=1}$ for an $n$-dimensional Hilbert space, and for every $\epsilon \gt 0$, there is a subset $I \subset {1,2,\ldots,m}$ with $|I| \geq (1-\epsilon)n$ so that $(f_{i})_{i \in I}$ is a Riesz basis for its span with Riesz basis constant a function of $\epsilon$, the frame bounds, and $(||f_{i}||)^{m}_{i=1}$, 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.