no. 1
On the angles between certain arithmetically defined subspaces of 𝐂 n
Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 175-185.

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·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·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.

@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},
     pages = {175--185},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {1},
     year = {1987},
     doi = {10.5802/aif.1081},
     zbl = {0611.15003},
     mrnumber = {89h:11022},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1081/}
}
TY  - JOUR
AU  - Brooks, Robert
TI  - On the angles between certain arithmetically defined subspaces of ${\bf C}^n$
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 175
EP  - 185
VL  - 37
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1081/
DO  - 10.5802/aif.1081
LA  - en
ID  - AIF_1987__37_1_175_0
ER  - 
%0 Journal Article
%A Brooks, Robert
%T On the angles between certain arithmetically defined subspaces of ${\bf C}^n$
%J Annales de l'Institut Fourier
%D 1987
%P 175-185
%V 37
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1081/
%R 10.5802/aif.1081
%G en
%F 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) no. 1, pp. 175-185. doi : 10.5802/aif.1081. https://aif.centre-mersenne.org/articles/10.5802/aif.1081/

[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

Cité par Sources :