Fibonacci representations of sequences in Hilbert spaces

Abstract

Dynamical sampling deals with frames of the form $\{T^n\phi {\}}_{n=0}^{\infty }$, where $T\in B(\mathcal{H})$ belongs to certain classes of linear operators and $\phi \in \mathcal{H}$. 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 $n⩾1$ 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.