Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case
[Polynômes orthogonaux avec coefficients de récurrence périodiquement modulés dans le cas bloc de Jordan]
Świderski, Grzegorz1 ; Trojan, Bartosz2
1 Mathematical Institute University of Wrocław pl. Grunwaldzki 2/4 50-384 Wrocław (Poland)
2 Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-696 Warszawa (Poland)
Annales de l'Institut Fourier, Online first, 81 p.

Nous étudions des polynômes orthogonaux avec coefficients de récurrence périodiquement modulés, lorsque 0 est une véritable extrémité du spectre de la matrice de Jacobi périodique correspondante. En particulier, nous montrons que leur mesure d’orthogonalité est purement absolument continue sur une demi-droite réelle et purement discrète sur son complémentaire. De plus, nous fournissons une formule constructive pour la densité, en termes de déterminants de Turán. Nous déterminons aussi le comportement asymptotique exact des polynômes orthogonaux. Enfin, nous étudions les limites d’échelle du noyau de Christoffel–Darboux.

We study orthogonal polynomials with periodically modulated recurrence coefficients when 0 lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel–Darboux kernel.

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DOI : 10.5802/aif.3624
Classification : 42C05, 47B36
Keywords: Orthogonal polynomials, asymptotics, Turán determinants, Christoffel functions, scaling limits
Mot clés : Polynômes orthogonaux, asymptotiques, déterminants de Turán, fonctions de Christoffel, limites d’échelle
Świderski, Grzegorz 1 ; Trojan, Bartosz 2

1 Mathematical Institute University of Wrocław pl. Grunwaldzki 2/4 50-384 Wrocław (Poland)
2 Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-696 Warszawa (Poland) 
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Świderski, Grzegorz; Trojan, Bartosz. Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case. Annales de l'Institut Fourier, Online first, 81 p.

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