Geodesic
An asymptotic formula for polynomials orthonormal with respect to a varying weight. II
Sbornik. Mathematics, Tome 205 (2014) no. 9, pp. 1334-1356 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight $e^{-2nQ(x)}p_g(x)/\sqrt{\prod_{j=1}^{2p}(x-e_j)}$ coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value problem on the relevant hyperelliptic Riemann surface of genus $g=p-1$. Here $e_1, $Q(x)=x^{2m}+\dotsb$ is a monic polynomial of even degree $2m$ and $p_g$ is a certain auxiliary polynomial of degree $p-1$. Bibliography: 23 titles.
Keywords: varying weight, strong asymptotics
Mots-clés : orthonormal polynomials, uniform distributions. 
@article{SM_2014_205_9_a4,
     author = {A. V. Komlov and S. P. Suetin},
     title = {An asymptotic formula for polynomials orthonormal with respect to a~varying {weight.~II}},
     journal = {Sbornik. Mathematics},
     pages = {1334--1356},
     year = {2014},
     volume = {205},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_9_a4/}
}
TY  - JOUR
AU  - A. V. Komlov
AU  - S. P. Suetin
TI  - An asymptotic formula for polynomials orthonormal with respect to a varying weight. II
JO  - Sbornik. Mathematics
PY  - 2014
SP  - 1334
EP  - 1356
VL  - 205
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2014_205_9_a4/
LA  - en
ID  - SM_2014_205_9_a4
ER  - 
%0 Journal Article
%A A. V. Komlov
%A S. P. Suetin
%T An asymptotic formula for polynomials orthonormal with respect to a varying weight. II
%J Sbornik. Mathematics
%D 2014
%P 1334-1356
%V 205
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2014_205_9_a4/
%G en
%F SM_2014_205_9_a4
A. V. Komlov; S. P. Suetin. An asymptotic formula for polynomials orthonormal with respect to a varying weight. II. Sbornik. Mathematics, Tome 205 (2014) no. 9, pp. 1334-1356. http://geodesic.mathdoc.fr/item/SM_2014_205_9_a4/

[1] A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 2012 (2012), 139–159 | MR | Zbl

[2] A. V. Komlov, S. P. Suetin, “Widom's formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight”, Russian Math. Surveys, 67:1 (2012), 183–185 | DOI | DOI | MR | Zbl

[3] A. A. Gonchar, E. A. Rakhmanov, “On the convergence of simultaneous Padé approximants for systems of functions of Markov type”, Proc. Steklov Inst. Math., 157 (1983), 31–50 | MR | Zbl

[4] P. D. Lax, C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. I”, Comm. Pure Appl. Math., 36:3 (1983), 253–290 | DOI | MR | Zbl

[5] P. D. Lax, C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. II”, Comm. Pure Appl. Math., 36:5 (1983), 571–593 | DOI | MR | Zbl

[6] P. D. Lax, C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. III”, Comm. Pure Appl. Math., 36:6 (1983), 809–829 | DOI | MR | Zbl

[7] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials”, Math. USSR-Sb., 53:1 (1986), 119–130 | DOI | MR | Zbl

[8] E. A. Rakhmanov, “On the asymptotic properties of polynomials that are orthogonal on the real axis”, Soviet Math. Dokl., 24 (1981), 505–507 | MR | Zbl

[9] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, X. Zhou, “Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory”, Comm. Pure Appl. Math., 52:11 (1999), 1335–1425 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[10] P. Deift, Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lect. Notes Math., 3, New York Univ., Courant Inst. Math. Sci., New York, NY; Amer. Math. Soc., Providence, RI, 1999, viii+273 pp. | MR | Zbl

[11] H. Widom, “Extremal polynomials associated with a system of curves in the complex plane”, Advances in Math., 3:2 (1969), 127–232 | DOI | MR | Zbl

[12] J. Nuttall, “Padé polynomial asymptotics from a singular integral equation”, Constr. Approx., 6:2 (1990), 157–166 | DOI | MR | Zbl

[13] S. P. Suetin, “Uniform convergence of Padé diagonal approximants for hyperelliptic functions”, Sb. Math., 191:9 (2000), 1339–1373 | DOI | DOI | MR | Zbl

[14] S. P. Suetin, “Comparative asymptotic behavior of solutions and trace formulas for a class of difference equations”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S96–S137 | DOI | DOI | MR | Zbl

[15] A. I. Aptekarev, V. Van Assshe, S. P. Suetin, “Skalyarnaya zadacha Rimana i silnaya asimptotika approksimatsii Pade i ortogonalnykh mnogochlenov”, Preprinty IPM, 2001, 026

[16] N. I. Ahiezer, “Orthogonal polynomials on several intervals”, Soviet Math. Dokl., 1 (1960), 989–992 | MR | Zbl

[17] N. I. Ahiezer, “Continuous analogues of orthogonal polynomials on a system of intervals”, Soviet Math. Dokl., 2 (1961), 1409–1412 | MR | Zbl

[18] F. Peherstorfer, “Zeros of polynomials orthogonal on several intervals”, Int. Math. Res. Not., 2003, no. 7, 361–385 | DOI | MR | Zbl

[19] A. I. Aptekarev, W. Van Assche, “Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight”, J. Approx. Theory, 129:2 (2004), 129–166 | DOI | MR | Zbl

[20] G. Springer, Introduction to Riemann surfaces, Addison-Wesley, Reading, MA, 1957, viii+307 pp. | MR | MR | Zbl | Zbl

[21] E. I. Zverovich, “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces”, Russian Math. Surveys, 26:1 (1971), 117–192 | DOI | MR | Zbl

[22] B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92 | DOI | MR | Zbl

[23] E. M. Chirka, “Rimanovy poverkhnosti”, Lekts. kursy NOTs, 1, MIAN, M., 2006, 3–105 | DOI