I'm studying frames from this paper. I have several question that I can't seem to find answer to.
- Why do we limit the definition on Hilbert space? What property do we need that other spaces don't have?
- Can I construct an equal norm tight frame from any tight frame? I know that I can make any frame an equal norm frame normalizing its vectors, but if I do that on a tight frame, it seems that I lose tightness
- At page 21, there is the formula $\sum_{i \in I}\left|\left<x, \phi_i\right>\right|^2 = \tilde\Phi\tilde\Phi^*\lVert x\rVert ^2 $, where x is a vector, $\phi_i$ are the frame vectors and $\tilde\Phi$ is the dual frame. It seems to me that the left side is a scalar, since it's a sum of scalar product, meanwhile the right side is a matrix, therefore they can't be equal. Am I wrong?
- Lastly, at page 31, they give us Theorem 4.2, after introducing the notion of frame potential, which formula is $FP(\Phi = \{\phi_i\}_{i\in I} = \sum_{i,j \in I} \left|\left<\phi_i, \phi_j\right>\right|^2$, and say that if we have $m \leq n$ vectors in our frame, with n the dimension of the space, the minimum of the FP is n, but if I take orthonormal vector it should be m, and if I take the vector to be orthogonal with a smaller norm it's even less. Same problem for the next point of the theorem.