\(\{x_k \}_k\) is a frame for a Hilbert space \(H\)

if \(A||x^2|| \leq \sum_k | \langle x, l_k \rangle | ^2 \leq B ||x||^2\)

\(\exists A,B\) such that \(0 < A \leq B < \infty\)

\(A = \lambda_{min} (\Phi \Phi^H)\)

\(B = \lambda_{max} (\Phi \Phi^H)\)

Synthesis Operator

\(\Phi x = \sum_k x_k \phi_k\)

Analysis Operator

\(\Phi^H x)_k = \langle x, \phi_k \rangle\)

#+end_analysis

#+begin_definition Tight Frame

\(\{x_k}_k\}\) is called a tight frame when

\(A = B = \lambda\)

Bi orthogonal basis

A set of vectors \(\{\phi_k^~\}_k\) is a bi-orthogonal basis to the basis \(\{\phi_k\}_k\) if

\[x = \sum_k \langlex, \phi_k^~ \rangle \phi_k\]

or

\[x = \Phi \Phi^~H x\]

• \(\Phi^~\) is a left inverse to \(\Phi\)

Canonical dual-frame

\(\widetilde{\Phi} = (\Phi \Phi^H)^{-1} \Phi\)