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Annales mathématiques du Québec

, Volume 39, Issue 1, pp 25–48 | Cite as

Valeur propre minimale d’une matrice de Toeplitz et d’un produit de matrices de Toeplitz

  • Philippe RambourEmail author
Article
  • 45 Downloads

Abstract

This paper is essentially devoted to the study of the minimal eigenvalue \(\lambda _{N,\alpha }\) of the Toepllitz matrice \(T_N(\varphi _{\alpha })\) where \(\varphi _{\alpha }(e^{i \theta })=\vert 1- e^{i \theta } \vert ^{2\alpha } c_{1}(e^{i \theta })\) with \(c_{1}\) a positive sufficiently smooth function and \(0<\alpha <\frac{1}{2}\). We obtain \(\lambda _{N,\alpha }\sim c_{\alpha }N^{-2\alpha }c_{1}(1)\) when N goes to infinity and we obtain bounds for \(c_{\alpha }\). To arrive at these results we give a theorem which suggests that the entries of \(T_N^{-1}(\varphi _{\alpha })\) and \(T_N (\varphi ^{-1}_\alpha )\) are closely related. If \(\alpha _1 + \alpha _2 > \frac{1}{2}\) we obtain the asymptotic of the minimal eigenvalue of \(T_N (\varphi _{\alpha _1}) T_N (\varphi _{\alpha _2})\).

Mots clef

Matrices de Toeplitz Produit de matrices de Toeplitz Valeur propre minimale Opérateurs à noyau 

Résumé

Nous donnons une expression asymptotique de la plus petite valeur propre \(\lambda _{N,\alpha }\) de la matrice \(T_N(\varphi _{\alpha })\)\(\varphi _{\alpha }(e^{i \theta })=\vert 1- e^{i \theta } \vert ^{2\alpha } c_{1}(e^{i \theta })\), avec \(c_{1}\) une fonction strictement positive suffisamment régulière et \(0<\alpha < \frac{1}{2}\). Nous obtenons \(\lambda _{N,\alpha }\sim c_{\alpha }N^{-2\alpha }c_{1}(1)\) et nous donnons un encadrement de \(c_{\alpha }\). Pour obtenir cet équivalent nous donnons et utilisons un théoréme qui relie les coefficients de \(T_N^{-1}(\varphi _\alpha )\) et ceux de \(T_N (\varphi ^{-1}_\alpha )\). Sous l’hypothse \(\alpha _1+\alpha _2 >\frac{1}{2}\) nous obtenons galement une expression asymptotique de la valeur propre minimale de \(T_N (\varphi _{\alpha _1}) T_N (\varphi _{\alpha _2}) \).

Mathematics Subject Classification

Primaire 47B35 Secondaire 47B34 

References

  1. 1.
    Böttcher, A.: The constants in the asymptotic formulas by Rambour and Seghier for inverses of Toeplitz matrices. Integr. Equ. Oper. Theory 50, 43–55 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)Google Scholar
  3. 3.
    Böttcher, A., Virtanen, J.: Norms of Toeplitz matrices with Fisher-Hartwig symbols. SIAM Matrix Anal. Appl. 29, 660–671 (2007)CrossRefGoogle Scholar
  4. 4.
    Böttcher, A., Widom, H.: From Toeplitz eigenvalues through Green’s kernels to higher-order Wirtinger-Sobolev inequalities. Oper. Theory Adv. Appl. 171, 73–87 (2006)CrossRefGoogle Scholar
  5. 5.
    Gohberg, I., Semencul, A.A.: The inversion of finite Toeplitz matrices and their continuous analogues [in Russian]. Mat. Issled. 7, 201–223 (1972)MathSciNetGoogle Scholar
  6. 6.
    Grenander, U., Szegö, G.: Toeplitz Forms and their Applications, 2nd edn. Chelsea, New York (1984)zbMATHGoogle Scholar
  7. 7.
    Kahane, J.P.: Séries de Fourier Absolument Convergentes. Springer, Berlin, Heidelberg, New York (1970)zbMATHGoogle Scholar
  8. 8.
    Landau, H.J.: Maximum entropy and the moment problem. Bull. (New Series) Am. Math. Soc. 16(1), 47–77 (1987)CrossRefzbMATHGoogle Scholar
  9. 9.
    Rambour, P.: Maximal eigenvalue and norm of a product of Toeplitz matrices. Stud. Part. Case Bull. Sci. Math. 137, 1072–1086 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rambour, P., Seghier, A.: Inversion des matrices de Toeplitz dont le symbole admet un zéro d’ordre rationnel positif, valeur propre minimale. Ann Faculté Sci Toulouse XXI(1), 173–211 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rambour, P., Seghier, A.: Formulas for the inverses of Toeplitz matrices with polynomially singular symbols. Integr. Equ. Oper. Theory 50, 83–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rambour, P., Seghier, A.: Théorèmes de trace de type Szegö dans le cas singulier. Bull. Sci. Math. 129, 149–174 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rambour, P., Seghier, A.: Inverse asymptotique des matrices de Toeplitz de symbole \((1-\cos \theta )^\alpha f_{1},\, \frac{-1}{2}{}\alpha < \frac{1}{2}\), et noyaux intégraux. Bull. Sci. Math. 134, 155–188 (2010)Google Scholar
  14. 14.
    Whittle, P.: Estimation and information in stationary time series. Ark. Mat. 2, 423–434 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zygmund, A.: Trigonometric Series, vol. 1. Cambridge University Press, Cambridge (1968)Google Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Université de Paris SudOrsay CedexFrance

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