Frames induced by the action of continuous powers of an operator

Abstract

We investigate systems of the form {A t g : g ∈ G, t ∈ [0, L]} where A ∈ B(H) is a normal operator in a separable Hilbert space H, G ⊂ H is a countable set, and L is a positive real number. Although the main goal of this work is to study the frame properties of {A t g : g ∈ G, t ∈ [0, L]}, as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.

Sampling the Flow of a Bandlimited Function

“…It was shown in [5,Theorem 5.5] that φ,L is a stable sampling set for some L > 0, if and only if φ,1 is a stable sampling set. Thus, for qualitative results, we will only consider the case of L = 1.…”

“…with A, B > 0; see Example 4.1 for a concrete construction. The emerging field of dynamical sampling investigates such phenomena in great generality (see, e.g., [1][2][3][4][5]). As follows from Example 4.1, the estimates (1.7) may hold with an arbitrary small sensor density.…”

“…In this section, we recall some of the results on dynamical sampling from [4,5] and adapt them for problems studied in this paper. For φ ∈ L 1 , consider the function…”

Continuous and discrete dynamical sampling

“…Further, since e tA is always invertible, the invertibility of A will not be necessary. Note that, as pointed out before, in general, it is not true that e tA =Ã t , and hence the results from [4] cannot be obtained immediately, although most of them will still be valid.…”

“…The problem of dynamical sampling for continuous time was originally posed for normal operators in [4]. There, since A is normal, the authors work with continuous powers {A t } t∈[0,L] defined though the Spectral Theorem.…”