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

Brooks, Robert

doi:10.5802/aif.1081

On the angles between certain arithmetically defined subspaces of

𝐂^{n}

Brooks, Robert

Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 175-185.

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.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1081

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     author = {Brooks, Robert},
     title = {On the angles between certain arithmetically defined subspaces of ${\bf C}^n$},
     journal = {Annales de l'Institut Fourier},
     pages = {175--185},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {1},
     year = {1987},
     doi = {10.5802/aif.1081},
     mrnumber = {89h:11022},
     zbl = {0611.15003},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1081/}
}

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Brooks, Robert. On the angles between certain arithmetically defined subspaces of ${\bf C}^n$. Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 175-185. doi : 10.5802/aif.1081. http://www.numdam.org/articles/10.5802/aif.1081/

Bibliographie
Cité par

[1] R. Brooks, The Bottom of the Spectrum of a Riemannian Covering, Crelles J., 357 (1985), 101-114. | MR | Zbl

[2] R. Brooks, The Spectral Geometry of the Apollonian Packing, Comm. P. Appl. Math., XXXVIII (1985), 357-366. | Zbl

[3] R. Brooks, The Spectral Geometry of a Tower of Coverings, J. Diff. Geom., 23 (1986), 97-107. | MR | Zbl

[4] Gelfand, Graev and Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, W.B. Saunders Co., 1969. | MR | Zbl

[5] H. Helson and D. Sarason, Past and Future, Math. Scand., 21 (1967), 5-16. | MR | Zbl

[6] A. Selberg, On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math, VIII (1965), 1-15. | MR | Zbl

[7] A. Weil, On Some Exponential Sums, Proc. Nat. Acad. Sci. USA, 34 (1948), 204-207. | MR | Zbl

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