Fibonacci representations of sequences in Hilbert spaces

Abstract: Dynamical sampling deals with frames of the form ${T^n\varphi}_{n=0}^\infty$, where $T \in B(\mathcal{H})$ belongs to certain classes of linear operators and $\varphi\in\mathcal{H}$. The purpose of this paper is to investigate a new representation, namely, Fibonacci representation of sequences ${f_n}{n=1}^\infty$ in a Hilbert space $\mathcal{H}$; having the form $f{n+2}=T(f_n+f_{n+1})$ for all $n\geqslant 1$ and a linear operator $T :\text{span}{f_n}{n=1}^\infty\to\text{span}{f_n}{n=1}^\infty$. We apply this kind of representations for complete sequences and frames. Finally, we present some properties of Fibonacci representation operators.