Fibonacci representations of sequences in Hilbert spaces

Abstract

Dynamical sampling deals with frames of the form {T n ϕ} ∞ n=0 , where T ∈ B(H) belongs to certain classes of linear operators and ϕ ∈ H. The purpose of this paper is to investigate a new representation, namely, Fibonacci representation of sequences {fn} ∞ n=1 in a Hilbert space H; having the form fn+2 = T (fn + fn+1) for all n 1 and a linear operator T : span{fn} ∞ n=1 → span{fn} ∞ n=1 . We apply this kind of representations for complete sequences and frames. Finally, we present some properties of Fibonacci representation operators.