Frame scalability in dynamical sampling

Let H be the finite dimensional Hilbert space R
ⁿ
or C
ⁿ
. We consider a subset G C H and a dynamical operator A : H → H. The collection F
_G
^L
(A) = {A
^j
g : g ∈ G, 0 ≤ j ≤ L(g)} is called a dynamical system, where L = (L
₁
, ... , L
_p
) is a vector of non-negative integers. Under certain conditions F
_G
^L
(A) is a frame for H; in such a case, we call this system a dynamical frame, generated by operator A and set G. A frame is a spanning set of a Hilbert space and a tight frame is a special case of a frame which admits a reconstruction formula similar to that of an orthonormal basis. Because of this simple formulation of reconstruction, tight frames are employed in many applications. Given a spanning set of vectors in H satisfying a certain property, one can manipulate the length of the vectors to obtain a tight frame. Such a spanning set is called a scalable frame. In this contribution, we study the relations between the operator A, the set G and the number of iterations L which ensure that the dynamical system F
_G
^L
(A) is a scalable frame.