A zero-sqrt(5)/ 2 law for cosine families
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ESTERLE, Jean | |
| dc.date.accessioned | 2024-04-04T03:18:32Z | |
| dc.date.available | 2024-04-04T03:18:32Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194415 | |
| dc.description.abstractEn | Let $a \in \R,$ and let $k(a)$ be the largest constant such that $sup\vert cos(na)-cos(nb)\vert < k(a)$ for $b\in \R$ implies that $b \in \pm a+2\pi\Z. $ We show thatif a cosine sequence $(C(n))_{n\in \Z}$ with values in a Banach algebra $A$ satisfies $sup_{n\ge 1}\Vert C(n) -cos(na).1_A\Vert < k(a),$ 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 $sup_{g\in G}\Vert C(g)-c(g)\Vert < {\sqrt 5\over 2}$ 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 $k(a)\le {3\over 2}.$ | |
| dc.language.iso | en | |
| dc.subject.en | cosine sequence | |
| dc.subject.en | cosine family | |
| dc.subject.en | cyclotomic polynomial | |
| dc.subject.en | Kronecker's theorem | |
| dc.title.en | A zero-sqrt(5)/ 2 law for cosine families | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 1505.06064 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-01147792 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01147792v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=ESTERLE,%20Jean&rft.genre=preprint |
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