A theorem contained in
Christensen, Ole (1995), "A Paley-Wiener theorem for frames." Proceedings of the American Mathematical Society, 123, 2199-2201.
states that
Let $\{x_n\}_{n\in\mathbb Z}$ be a frame for a Hilbert space $H$ with bounds $A$ and $B$. Let $\{y_n\}_{n \in\mathbb Z}\subset H$ be such that the inequality $\left\|\sum_ n a_n (x_n-y_n)\right\|^2\leq C \sum_n |a_n|^2 $ is valid with a constant $0< C<A$. Then, $\{y_n\}_{n\in\mathbb Z}$ is a frame with bounds $A'\ge (1-(\mbox{$\frac C A$})^{\frac 12})^2 $ and $B'\leq (1+(\mbox{$\frac C A$})^{\frac 12})^2 $. If $\{x_n\}_{n \in\mathbb Z}$ is a Riesz basis, then $\{y_n\}_{n\in\mathbb Z}$ is also a Riesz basis.
This theorem allows to estimate the frames bounds of a frame by knowing the frame bounds of another frame. My question is maybe too naive, but why should we determine the estimates of frames bounds? Does someone know an application where these estimates are important? Thanks!