Currently investigating this question :
Let $s = \{s_1, s_2, ..., s_M\}$ be a set of $M \geq N$ vectors in $\mathbb{C}^N$ and let $S$ be a $N \times M$ matrix s.t. $S = [s_1, s_2, ..., s_M]$, what are sufficient conditions on S s.t. $s = \{s_m\}$ is a frame for $\mathbb{C}^N$, what are the frame bounds and what is the frame operator?
My input :
Since $\mathbb{C}^N$ is equipped with the usual inner product, we can state as per frame definition that
$A_s \cdot \left\| x \right\|_{\mathcal{H}}^2 \leq \sum_{n = 1}^M | \langle x, s_n \rangle |^2 \leq B_s \cdot \left\| x \right\|^2$
I can take the inner term and rewrite that to
$\sum_{n = 1}^M | \langle x, s_n \rangle |^2 = \sum_{n = 1}^M \left| \sum_{j = 1}^N x_j \overline{S_{j, n}} \right |^2$
If I am not mistaken, this should be equal to $\left\| \Theta x \right\|_{l^2}^2$ with $\Theta$ being the analysis operator. We know then that $S = \Theta^*$ (which is the synthesis operator) and that $S$ must have rank $M$ s.t. $s = \{s_m\}$ is a frame.
The bounds are then formally given as :
- $A_s \leq \frac{\left\| \Theta x \right\|_{l^2}^2}{\left\| x \right\|^2}$
- $B_s \geq \frac{\left\| \Theta x \right\|_{l^2}^2}{\left\| x \right\|^2}$
and i think that I read in this book Frames For Undergraduates that $A_s = \lambda_{min}$ and $B_s = \lambda_{max}$ with $\lambda_i$ being eigenvalues of of $SS^*$.
For the frame operator $F = SS^*$ I am really stuck. I know that $F$ should be self-adjoint and positive, so in theory, I should be able to write according to SVD that
$F = U^* \Lambda U$
However, I am not sure. Does anybody mind quickly going through that and suggesting some alternative for $F$ ? Or a more precise statement?
Edit :
F looks as the following and represents a matrix full of outer product terms where the diagonals are note squared norms of the frame elements.
$ \begin{align*} F = S \times S^* &= \begin{bmatrix} s_{1, 1} & s_{2, 1} & ... & s_{M, 1} \\ s_{1, 2} & s_{2, 2} & ... & s_{M, 2} \\ \vdots & \vdots & ... & \vdots \\ s_{1, N} & s_{2, N} & ... & s_{M, N} \end{bmatrix} \cdot \begin{bmatrix} \overline{s_{1, 1}} & \overline{s_{2, 1}} & ... & \overline{s_{M, 1}} \\ \overline{s_{1, 2}} & \overline{s_{2, 2}} & ... & \overline{s_{M, 2}} \\ \vdots & \vdots & ... & \vdots \\ \overline{s_{1, N}} & \overline{s_{2, N}} & ... & \overline{s_{M, N}} \end{bmatrix} \\ &= \begin{bmatrix} \sum_{j = 1}^M s_{j, 1} \overline{s_{j, 1}} & \sum_{j = 1}^M s_{j, 1} \overline{s_{j, 2}} & ... & \sum_{j = 1}^M s_{j, 1} \overline{s_{j, N}} \\ \sum_{j = 1}^M s_{j, 2} \overline{s_{j, 1}} & \sum_{j = 1}^M s_{j, 2} \overline{s_{j, 2}} & ... & \sum_{j = 1}^M s_{j, 2} \overline{s_{j, N}} \\ \vdots & \vdots & ... & \vdots \\ \sum_{j = 1}^M s_{j, N} \overline{s_{j, 1}} & \sum_{j = 1}^M s_{j, N} \overline{s_{j, 2}} & ... & \sum_{j = 1}^M s_{j, N} \overline{s_{j, N}} \end{bmatrix} \end{align*} $