Geodesic
On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szegö's condition
Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 149-167 
E. A. Rakhmanov. On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szegö's condition. Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 149-167. http://geodesic.mathdoc.fr/item/SM_1987_58_1_a8/
@article{SM_1987_58_1_a8,
     author = {E. A. Rakhmanov},
     title = {On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the {Szeg\"o's} condition},
     journal = {Sbornik. Mathematics},
     pages = {149--167},
     year = {1987},
     volume = {58},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1987_58_1_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author considers asymptotic properties of polynomials $\varphi_n(z)$, orthonormal on the unit circle $\Gamma$, with weights $f(z)$ that do not satisfy Szegö's condition. It is shown, in particular, that if $f(z)$ satisfies a Dini–Lipschitz condition, then $\lim_{n\to\infty}|\varphi_n(z)|=f(z)^{-1/2}$ uniformly on each set $\gamma\subset\Gamma$ on which $f$ has a positive lower bound. Bibliography: 9 titles.

[1] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[2] Geronimus Ya. L., Mnogochleny, ortogonalnye na okruzhnosti i otrezke, Fizmatgiz, M., 1958 | Zbl

[3] Rakhmanov E. A., “Ob asimptotike otnosheniya ortogonalnykh mnogochlenov”, Matem. sb., 103(145) (1977), 237–252 | Zbl

[4] Rakhmanov E. A., “Ob asimptotike otnosheniya ortogonalnykh mnogochlenov II”, Matem. sb., 118(160) (1982), 104–117 | MR | Zbl

[5] Gonchar A. A., “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matem. sb., 97(139) (1975), 607–629 | Zbl

[6] Lopes G., “Ob asimptotike otnosheniya ortogonalnykh mnogochlenov i skhodimosti mnogotochechnykh approksimatsii Pade”, Matem. sb., 128(170) (1985), 216–229 | MR

[7] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1972 | MR

[8] Badkov V. M., “Asimptoticheskoe povedenie ortogonalnykh mnogochlenov”, Matem. sb., 109(151) (1979), 46–59 | MR | Zbl

[9] A. Mate, P. Nevai, V. Totik, “Asymptotics for the ratio of leading coefficient of orthogonal polynomials on the unit circle”, Constructive Approximation I, 1985, 63–69 | MR | Zbl