Asymptotic inversion of Toeplitz matrices with one singularity in the symbol

Rambour, Philippe; Seghier, Abdellatif

doi:10.1016/j.crma.2009.02.024

Mathematical Analysis

Asymptotic inversion of Toeplitz matrices with one singularity in the symbol

Presented by: Jean-Pierre Kahane

Rambour, Philippe ¹; Seghier, Abdellatif ¹

¹ Université de Paris Sud, bâtiment 425, 91405 Orsay cedex, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 489-494.

Abstract
Résumé

We consider the function $f_{α}$ defined by $f_{α} (θ) = {| 1 - e^{i θ} |}^{2 α} f_{1} (e^{i θ})$ with $f_{1}$ a regular strictly positive function and α a real number with $- \frac{1}{2} < α$ . For such a number α we compute the inverse $T_{N} {(f_{α})}^{- 1}$ of the Toeplitz matrix $T_{N} (f_{α})$ and we obtain the asymptotic behaviour of the entries of this matrix when N goes to infinity. This inversion allows us to obtain two new families of kernels, $H_{α}$ for $- \frac{1}{2} < α < 0$ , and $G_{α}$ for $α > 0$ . We obtain also an asymptotic expansion of the coefficients of the orthogonal polynomials associated to the function $f_{α}$ and we give an answer to a question of H. Kesten (1961).

Considérons la fonction $f_{α} (θ) = {| 1 - e^{i θ} |}^{2 α} f_{1} (e^{i θ})$ où $f_{1}$ est une fonction régulière strictement positive et α un nombre réel tel que $- \frac{1}{2} < α$ . Pour un tel α nous calculons l'inverse $T_{N} {(f_{α})}^{- 1}$ de la matrice de Toeplitz $T_{N} (f_{α})$ et nous obtenons le comportement asymptotique des coefficients de cet inverse quand N tend vers l'infini. Ceci nous permet de mettre en évidence deux nouvelles familles de noyaux $H_{α}$ pour $- \frac{1}{2} < α < 0$ , et $G_{α}$ pour $α > 0$ . Nous obtenons également un développement asymptotique des coefficients des polynômes orthogonaux associés au poids $f_{α}$ . Pour $α = \frac{1}{2}$ nous répondons à une question énoncée par H. Kesten (1961).

Received: 2007-03-28
Accepted: 2009-02-16
Published online: 2009-03-31

DOI: 10.1016/j.crma.2009.02.024

Author's affiliations:

Rambour, Philippe ¹; Seghier, Abdellatif ¹

¹ Université de Paris Sud, bâtiment 425, 91405 Orsay cedex, France

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     author = {Rambour, Philippe and Seghier, Abdellatif},
     title = {Asymptotic inversion of {Toeplitz} matrices with one singularity in the symbol},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {489--494},
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     volume = {347},
     number = {9-10},
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     doi = {10.1016/j.crma.2009.02.024},
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Rambour, Philippe; Seghier, Abdellatif. Asymptotic inversion of Toeplitz matrices with one singularity in the symbol. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 489-494. doi : 10.1016/j.crma.2009.02.024. http://www.numdam.org/articles/10.1016/j.crma.2009.02.024/

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