A ZERO- LAW FOR COSINE FAMILIES
| Autor: | Jean Esterle |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
General Mathematics
010102 general mathematics Zero (complex analysis) 01 natural sciences Combinatorics symbols.namesake 0103 physical sciences symbols Trigonometric functions Kronecker's theorem 010307 mathematical physics 0101 mathematics Abelian group Cyclotomic polynomial Banach *-algebra Real number Mathematics |
| Zdroj: | Journal of the Australian Mathematical Society. 104:195-217 |
| ISSN: | 1446-8107 1446-7887 |
| DOI: | 10.1017/s1446788717000118 |
| Popis: | Let$a\in \mathbb{R}$, and let$k(a)$be the largest constant such that$\sup |\text{cos}(na)-\cos (nb)|for$b\in \mathbb{R}$implies that$b\in \pm a+2\unicode[STIX]{x1D70B}\mathbb{Z}$. We show that if a cosine sequence$(C(n))_{n\in \mathbb{Z}}$with values in a Banach algebra$A$satisfies$\sup _{n\geq 1}\Vert C(n)-\cos (na).1_{A}\Vert , then$C(n)=\cos (na).1_{A}$for$n\in \mathbb{Z}$. Since$\!\sqrt{5}/2\leq k(a)\leq 8/3\!\sqrt{3}$for every$a\in \mathbb{R}$, this shows that if some cosine family$(C(g))_{g\in G}$over an abelian group$G$in a Banach algebra satisfies$\sup _{g\in G}\Vert C(g)-c(g)\Vert for some scalar cosine family$(c(g))_{g\in G}$, then$C(g)=c(g)$for$g\in G$, and the constant$\!\sqrt{5}/2$is optimal. We also describe the set of all real numbers$a\in [0,\unicode[STIX]{x1D70B}]$satisfying$k(a)\leq \frac{3}{2}$. |
| Databáze: | OpenAIRE |
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