Geodesic
The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$
Sbornik. Mathematics, Tome 209 (2018) no. 7, pp. 985-1018 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the polynomials $\{\varphi_n(z,w)\}$ orthogonal on the circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$ and show that $\|\varphi_n(e^{i\theta},w)\|_{L^\infty(\mathbb{T})}$ can grow in $n$ at a certain rate. Bibliography: 21 titles.
Keywords: polynomials orthogonal on the circle
Mots-clés : Steklov's conjecture. 
@article{SM_2018_209_7_a3,
     author = {S. A. Denisov},
     title = {The growth of polynomials orthogonal on the unit circle with respect to a~weight~$w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$},
     journal = {Sbornik. Mathematics},
     pages = {985--1018},
     year = {2018},
     volume = {209},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_7_a3/}
}
TY  - JOUR
AU  - S. A. Denisov
TI  - The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 985
EP  - 1018
VL  - 209
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_7_a3/
LA  - en
ID  - SM_2018_209_7_a3
ER  - 
%0 Journal Article
%A S. A. Denisov
%T The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$
%J Sbornik. Mathematics
%D 2018
%P 985-1018
%V 209
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2018_209_7_a3/
%G en
%F SM_2018_209_7_a3
S. A. Denisov. The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty(\mathbb{T})$. Sbornik. Mathematics, Tome 209 (2018) no. 7, pp. 985-1018. http://geodesic.mathdoc.fr/item/SM_2018_209_7_a3/

[1] M. U. Ambroladze, “On the possible rate of growth of polynomials orthogonal with a continuous positive weight”, Math. USSR-Sb., 72:2 (1992), 311–331 | DOI | MR | Zbl

[2] A. Aptekarev, S. Denisov, D. Tulyakov, “On a problem by Steklov”, J. Amer. Math. Soc., 29:4 (2016), 1117–1165 | DOI | MR | Zbl

[3] S. Bernstein, “Sur les polynomes orthogonaux relatifs à un segment fini”, J. Math. (9), 9 (1930), 127–177 ; II, 10 (1931), 219–286 | Zbl | Zbl

[4] S. Denisov, “Remark on the formula by Rakhmanov and Steklov's conjecture”, J. Approx. Theory, 205 (2016), 102–113 | DOI | MR | Zbl

[5] G. B. Folland, Real analysis. Modern techniques and their applications, Pure Appl. Math. (N. Y.), Wiley-Interscience Publ. John Wiley Sons, Inc., New York, 1999, xvi+386 pp. | MR | Zbl

[6] Ya. L. Geronimus, Polynomials orthogonal on a circle and interval, International Series of Monographs on Pure and Applied Mathematics, 18, Pergamon Press, New York–Oxford–London–Paris, 1960, ix+210 pp. | MR | MR | Zbl | Zbl

[7] Ja. L. Geronimus, “Some estimates of orthogonal polynomials and on Steklov's problem”, Soviet Math. Dokl., 18 (1978), 1176–1179 | MR | Zbl

[8] Ya. L. Geronimus, “O zavisimosti mezhdu poryadkom rosta ortonormalnykh mnogochlenov i kharakterom oblozheniya”, Matem. sb., 61(103):1 (1963), 65–79 | MR | Zbl

[9] J. L. Geronimus, “On Steklov's conjecture”, Soviet Math. Dokl., 3 (1962), 74–76 | MR | Zbl

[10] B. L. Golinskii, “The V. A. Steklov problem in the theory of orthogonal polynomials”, Math. Notes, 15:1 (1974), 13–19 | DOI | MR | Zbl

[11] B. Hollenbeck, I. E. Verbitsky, “Best constants for the Riesz projection”, J. Funct. Anal., 175:2 (2000), 370–392 | DOI | MR | Zbl

[12] A. Máté, P. Nevai, V. Totik, “Extensions of Szegő's theory of orthogonal polynomials. II”, Constr. Approx., 3:1 (1987), 51–72 ; III, 73–96 | DOI | MR | Zbl | DOI | Zbl

[13] P. Nevai, J. Zhang, V. Totik, “Orthogonal polynomials: their growth relative to their sums”, J. Approx. Theory, 67:2 (1991), 215–234 | DOI | MR | Zbl

[14] E. A. Rakhmanov, “On Steklov's conjecture in the theory of orthogonal polynomials”, Math. USSR-Sb., 36:4 (1980), 549–575 | DOI | MR | Zbl

[15] E. A. Rakhmanov, “Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero”, Math. USSR-Sb., 42:2 (1982), 237–263 | DOI | MR | Zbl

[16] B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, Amer. Math. Soc. Colloq. Publ., 54, Part 1, Amer. Math. Soc., Providence, RI, 2005, xxvi+466 pp. ; Part 2. Spectral theory, Amer. Math. Soc. Colloq. Publ., 54, Part 2, i–xxii and 467–1044 | DOI | MR | Zbl | DOI | MR | Zbl

[17] W. Stekloff (V. Steklov), “Une méthode de la solution du problème de développement des fonctions en séries de polynomes de Tchébychef indépendante de la théorie de fermeture. I”, Izvѣstiya Rossiiskoi Akademii Nauk'. VI seriya, 15 (1921), 281–302 ; II, 303–326 | Zbl

[18] P. K. Suetin, “V. A. Steklov's problem in the theory of orthogonal polynomials”, J. Soviet Math., 12:6 (1979), 631–682 | DOI | MR | Zbl

[19] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975, xiii+432 pp. | MR | Zbl | Zbl

[20] R. Szwarc, “Uniform subexponential growth of orthogonal polynomials”, J. Approx. Theory, 81:2 (1995), 296–302 | DOI | MR | Zbl

[21] A. Zygmund, Trigonometric series, v. 1, 2, Cambridge Math. Lib., 3rd ed., Cambridge Univ. Press, Cambridge, 2002, xiv+383 pp., viii+364 pp. | MR | MR | Zbl | Zbl