Frame approximation with bounded coefficients

Abstract

Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this illconditioning is that the coefficients of a frame approximation can grow large. In this paper we resolve this issue by introducing two methods for frame approximation that possess bounded coefficients. As we show, these methods typically lead to little or no deterioration in the approximation accuracy, but successfully avoid the large coefficients inherent to previous approaches, thus making them attractive where large coefficients are undesirable. We also present theoretical analysis to support these conclusions.