Scalability of frames generated by dynamical operators

2016/08/19 by Aceska, Roza, Kim, Yeon Hyang · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1608.05622

Abstract

Let A be an operator on a separable Hilbert space \cH, and let G ⊂ \cH. It is known that - under appropriate conditions on A and G - the set of iterations F_G(A)= \A^j \gbf | \gbf ∈ G, 0 ≤ j ≤ L(\gbf) \ is a frame for \cH. We call F_G(A) a dynamical frame for \cH, and explore further its properties; in particular, we show that the canonical dual frame of F_G(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 F_G(A) is a scalable frame. We give a general statement on frame scalability, We and study in detail the case when A is a normal operator, utilizing the unitary diagonalization in finite dimensions. In addition, we answer the question of when F_G(A) is a scalable frame in several special cases involving block-diagonal and companion operators.

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