LOCAL THEORY OF FRAMES AND SCHAUDER BASES FOR HILBERT SPACE

摘要

We develope a local theory for frames on finite-dimensional Hilbert spaces. We show that for every frame (f_i)_i=1~m for an n-dimensional Hilbert space, and for every implied by > 0, there is a subset I is contained in {1,2,...,m} with |I| >= (1- implied by )n so that (f_i)_i implied by I is a Riesz basis for its span with Riesz basis constant a function of ∈, 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 are results are best possible, and that all parameters are necessary.