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.

由于其灵活性，希尔伯特空间的框架是逼近方案中有吸引力的替代方案，可解决那些无法确定基础甚至不可行的问题。但是，使用帧计算最佳近似值可能会很困难，因为它需要求解病态线性系统。这种不良情况的一个后果是，这种帧近似的系数可能会变大。在本文中，我们通过介绍两种具有有界系数的帧逼近方法来解决此问题。正如我们所展示的，这些方法通常不会导致近似精度降低或几乎不会降低，但是成功避免了先前方法固有的大系数，从而使它们在不希望使用大系数的情况下具有吸引力。