Scalability of Frames Generated by Dynamical Operators

• Main Authors: Roza Aceska, Yeon H. Kim
• Format: Article
• Language: English
• Published: Frontiers Media S.A. 2017-11-01
• Series: Frontiers in Applied Mathematics and Statistics
• Subjects: dynamical sampling; frames; scalable frames; iterative actions of operators
• Online Access: http://journal.frontiersin.org/article/10.3389/fams.2017.00022/full

Let ℍ be a separable Hilbert space, let G ⊂ ℍ, and let A be an operator on ℍ. Under appropriate conditions on A and G, it is known that the set of iterations FG(A)={Ajg|g∈G,0≤j≤L(g)} is a frame for ℍ. We call FG(A) a dynamical frame for ℍ, and explore further its properties; in particular, we show that the canonical dual frame of FG(A) also has an iterative set structure. We explore the relations between the operator A, the set G and the number of iterations L which ensure that the system FG(A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when FG(A) is a scalable frame in several special cases involving block-diagonal and companion operators.