A zero-sqrt(5)/ 2 law for cosine families

TL;DR

This paper establishes a sharp bound, approximately 1.118, for the stability of cosine families in Banach algebras, showing that deviations below this threshold guarantee the family coincides with a scalar cosine family.

Contribution

The paper proves a new optimal stability bound for cosine families in Banach algebras, extending classical results to a broader setting with precise constants.

Findings

The stability constant k(a) satisfies 0.866... ≤ k(a) ≤ 2.31 for all a.

If a cosine family deviates less than √5/2 from a scalar cosine family, they are identical.

The set of a in [0,π] with k(a) ≤ 3/2 is characterized.

Abstract

Let $a \in \R,$ and let $k(a)$ be the largest constant such that $sup\vert cos(na)-cos(nb)\vert \textless{} k(a)$ for $b\in \R$ implies that $b \in \pm a+2\pi\Z. $ We show that if a cosine sequence $(C(n))\_{n\in \Z}$ with values in a Banach algebra $A$ satisfies $sup\_{n\ge 1}\Vert C(n) -cos(na).1\_A\Vert \textless{} k(a),$ then $C(n)=cos(na)$ for $n\in \Z.$ Since ${\sqrt 5\over 2} \le k(a) \le {8\over 3\sqrt 3}$ for every $a \in \R,$ this shows that if some cosine family $(C(g))\_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies $sup\_{g\in G}\Vert C(g)-c(g)\Vert \textless{} {\sqrt 5\over 2}$ for some scalar cosine family $(c(g))\_{g\in G},$ then $C(g)=c(g)$ for $g\in G,$ and the constant ${\sqrt 5\over 2}$ is optimal. We also describe the set of all real numbers $a \in [0,\pi]$ satisfying $k(a)\le {3\over 2}.$