On the angles between certain arithmetically defined subspaces of ${\bf C}^n$

Brooks, Robert

On the angles between certain arithmetically defined subspaces of

𝐂^{n}

Brooks, Robert

Annales de l'Institut Fourier, Tome 37 (1987), p. 175-185 / Harvested from Numdam

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Résumé
Abstract

Pour ${v_{i}}$ et ${w_{j}}$ deux familles de bases de $C^{n}$ , et $θ$ un nombre fixe, nous considérons $V^{n}$ et $W^{n}$ deux sous-espaces engendrés par $[θ \cdot n]$ vecteurs de ${v_{i}}$ et ${w_{j}}$ respectivement. Nous étudions l’angle entre $V^{n}$ et $W^{n}$ quand $n$ tend vers l’infini. Nous démontrons que, quand ${v_{i}}$ et ${w_{j}}$ sont présents dans certaines familles définies arithmétiquement, l’angle entre $V^{n}$ et $W^{n}$ peut soit tendre vers 0, soit être minoré par une constante strictement positive. Ce comportement dépend d’un problème de valeur propre associé.

If ${v_{i}}$ and ${w_{j}}$ are two families of unitary bases for $C^{n}$ , and $θ$ is a fixed number, let $V^{n}$ and $W^{n}$ be subspaces of $C^{n}$ spanned by $[θ \cdot n]$ vectors in ${v_{i}}$ and ${w_{j}}$ respectively. We study the angle between $V^{n}$ and $W^{n}$ as $n$ goes to infinity. We show that when ${v_{i}}$ and ${w_{j}}$ arise in certain arithmetically defined families, the angles between $V^{n}$ and $W^{n}$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Publié le : 1987-01-01

MR 89h:11022 | Zbl 0611.15003

DOI : https://doi.org/10.5802/aif.1081

@article{AIF_1987__37_1_175_0,
     author = {Brooks, Robert},
     title = {On the angles between certain arithmetically defined subspaces of ${\bf C}^n$},
     journal = {Annales de l'Institut Fourier},
     volume = {37},
     year = {1987},
     pages = {175-185},
     doi = {10.5802/aif.1081},
     mrnumber = {89h:11022},
     zbl = {0611.15003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1987__37_1_175_0}
}

Brooks, Robert. On the angles between certain arithmetically defined subspaces of ${\bf C}^n$. Annales de l'Institut Fourier, Tome 37 (1987) pp. 175-185. doi : 10.5802/aif.1081. http://gdmltest.u-ga.fr/item/AIF_1987__37_1_175_0/

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