Fibonacci representations of sequences in Hilbert spaces

TL;DR

This paper introduces Fibonacci-based sequence representations in Hilbert spaces, exploring their properties, applications to frames, and the characteristics of associated operators, expanding the understanding of sequence structures in functional analysis.

Contribution

It proposes a novel Fibonacci representation for sequences in Hilbert spaces and analyzes its properties and applications to frames and operators.

Findings

Fibonacci representations can describe complete sequences and frames.

Properties of Fibonacci representation operators are characterized.

Applications to frame theory are demonstrated.

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.