On démontre qu’un sous-espace d’un espace de Hilbert de fonctions holomorphes est complètement défini par ses distances aux noyaux reproduisants. Une méthode simple est proposée pour localiser les zéros simultanés d’un sous-espace de l’espace de Hardy. À titre d’illustration on montre une famille de disques du plan complexe sans zéro de la fonction de Riemann.
It is proved that a subspace of a holomorphic Hilbert space is completely determined by their distances to the reproducing kernels. A simple rule is established to localize common zeros of a subspace of the Hardy space of the unit disc. As an illustration we show a series of discs of the complex plan free of zeros of the Riemann -function.
@article{AIF_1995__45_1_143_0,
author = {Nikolski, Nikolai},
title = {Distance formulae and invariant subspaces, with an application to localization of zeros of the {Riemann} $\zeta $-function},
journal = {Annales de l'Institut Fourier},
pages = {143--159},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {45},
number = {1},
year = {1995},
doi = {10.5802/aif.1451},
mrnumber = {96c:47005},
zbl = {0816.30026},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1451/}
}
TY - JOUR AU - Nikolski, Nikolai TI - Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann $\zeta $-function JO - Annales de l'Institut Fourier PY - 1995 SP - 143 EP - 159 VL - 45 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1451/ DO - 10.5802/aif.1451 LA - en ID - AIF_1995__45_1_143_0 ER -
%0 Journal Article %A Nikolski, Nikolai %T Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann $\zeta $-function %J Annales de l'Institut Fourier %D 1995 %P 143-159 %V 45 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1451/ %R 10.5802/aif.1451 %G en %F AIF_1995__45_1_143_0
Nikolski, Nikolai. Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann $\zeta $-function. Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 143-159. doi : 10.5802/aif.1451. http://www.numdam.org/articles/10.5802/aif.1451/
[B] , A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. USA, 41, n° 5 (1995), 312-314. | MR | Zbl
[DSS] , and , Singular measures and domains not of Smirnov type, Duke Math. Journal, 33 (1966), 247-254. | MR | Zbl
[H] , A factorization theorem for square area-integrable analytic functions, J. reine angew. Math., 422 (1991), 45-68. | MR | Zbl
[He] , An invariant subspace of the Bergman space having the codimension two property, J. reine angew. Math., 443 (1993), 1-9. | Zbl
[Ho] , Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, NJ, 1962. | MR | Zbl
[K] , Closed ideals of the ring An, Functional Analysis and its applications, 6, n° (1972), 38-52. | MR | Zbl
[Ko] , A Beurling type theorem, Acta Math., 138 (1977), 265-293. | MR | Zbl
[KR] and , A Phragmén-Lindelöf theorem with applications to M(u, v) functions, Pacific J. Math., 43 (1972), 175-188. | MR | Zbl
[N] , Treatise on the shift operator, Springer-Verlag, Heidelberg, 1986.
[Ni] , Invitation aux techniques des espaces de Hardy, EDM, Université Bordeaux-1, 1992, p. 1-69.
[Nik] , Two problems on spectral synthesis, Lect. Notes Math., Springer-Verlag, 1043 (1984), 378-381.
[Ny] , On some groups and semi-groups of translations, Thesis, Uppsala, 1950.
[Sh] , Analytic functions smooth up to the boundary, Lect. Notes Math., v. 1312, Springer-Verlag, 1988. | MR | Zbl
[V] , On a biorthogonal function system related to the Riemann hypothesis, Algebra i Analiz (St. Petersburg Math. J.), to appear. | MR | Zbl
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