  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 ill-conditioning is that
the coefficients of such 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 in situations where large coefficients are undesirable. We also
present theoretical analysis to support these conclusions.
