Geodesic
Asymptotics of Orthogonal Polynomials Beyond the Scope of Szeg\H{o}'s Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 4, pp. 22-32.

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First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros; then there exists an outer function $\phi$, $\phi(0)=1$, such that $\|(B\phi)'\|\le C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.
Mots-clés : orthogonal polynomial, CMV matrix.
Keywords: extremal problem, Blaschke product 
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A. L. Vol'berg; F. Peherstorfer; P. M. Yuditskii. Asymptotics of Orthogonal Polynomials Beyond the Scope of Szeg\H{o}'s Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 4, pp. 22-32. http://geodesic.mathdoc.fr/item/FAA_2006_40_4_a2/

[1] E. Gluskin, M. Meyer, A. Pajor, “Zeros of analytic functions and norms of inverse matrices”, Israel J. Math., 87:1–3 (1994), 225–242 | DOI | MR | Zbl

[2] N. K. Nikolskii, “Condition numbers of large matrices and analytic capacities”, Algebra i analiz, 17:4 (2005), 125–180 | MR

[3] F. Peherstorfer, P. Yuditskii, “Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points”, Proc. Amer. Math. Soc., 129:11 (2001), 3213–3220 | DOI | MR | Zbl

[4] H. Queffelec, “Sur un théorème de Gluskin–Meyer–Pajor”, C. R. Acad. Sci. Paris. Sér. I Math., 317:2 (1993), 155–158 | MR | Zbl

[5] J. J. Scha̋ffer, “Norms and determinants of linear mappings”, Math. Z., 118 (1970), 331–339 | DOI | MR | Zbl

[6] I. V. Videnskii, N. A. Shirokov, “Ob odnoi ekstremalnoi zadache v algebre Vinera”, Algebra i analiz, 11:6 (1999), 122–138 | MR