Tight Frames and Dual Frame Pairs

Abstract

We have already highlighted the frame decomposition, which shows that a frame $\{f_{k}\}_{k=1}^{\infty }$ for a Hilbert space $\mathcal{H}$ leads to the decomposition

$$\displaystyle\begin{array}{rcl} f =\sum _{ k=1}^{\infty }\langle f,S^{-1}f_{ k}\rangle f_{k},\ \ \forall f \in \mathcal{H};& &{}\end{array}$$

(6.1)

here $S: \mathcal{H}\rightarrow \mathcal{H}$ denotes the frame operator. In practice, it is difficult to apply the general frame decomposition, due to the fact that we need to invert the frame operator. We have mentioned two ways to circumvent the problem. The first one is to restrict our attention to tight frames: as we have seen in Corollary 5.1.7, for a tight frame $\{f_{k}\}_{k=1}^{\infty }$ with frame bound A, the frame decomposition takes the much simpler form

$$\displaystyle\begin{array}{rcl} f = \frac{1} {A}\sum _{k=1}^{\infty }\langle f,f_{ k}\rangle f_{k},\ \forall f \in \mathcal{H}.& &{}\end{array}$$

(6.2)