Asymptotic expansions for polynomials orthogonal with respect to a complex non-constant weight function
HTML articles powered by AMS MathViewer
- by
A. Aptekarev and R. Khabibullin
Translated by: Michael Grinfeld - Trans. Moscow Math. Soc. 2007, 1-37
- DOI: https://doi.org/10.1090/S0077-1554-07-00167-7
- Published electronically: November 15, 2007
- PDF | Request permission
Abstract:
We consider a sequence of polynomials that are orthogonal with respect to a complex analytic weight function which depends on the index $n$ of the polynomial. For such polynomials we obtain an asymptotic expansion in $1/n$. As an example, we present the asymptotic expansion for Laguerre polynomials with a weight that depends on the index of the polynomial.References
- A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and the rate of rational approximation of analytic functions, Mat. Sb. (N.S.) 134(176) (1987), no. 3, 306–352, 447 (Russian); English transl., Math. USSR-Sb. 62 (1989), no. 2, 305–348. MR 922628, DOI 10.1070/SM1989v062n02ABEH003242
- A. I. Aptekarev, Sharp constants for rational approximations of analytic functions, Mat. Sb. 193 (2002), no. 1, 3–72 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 1-2, 1–72. MR 1906170, DOI 10.1070/SM2002v193n01ABEH000619
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and 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 (1999), no. 11, 1335–1425. MR 1702716, DOI 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
- Pavel Bleher and Alexander Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324, DOI 10.2307/121101
- L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), no. 1-2, 109–147. MR 1435193, DOI 10.1007/BF02180200
- Jinho Baik, Percy Deift, Ken T.-R. McLaughlin, Peter Miller, and Xin Zhou, Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1207–1250. MR 1926668, DOI 10.4310/ATMP.2001.v5.n6.a7
- Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178. MR 1682248, DOI 10.1090/S0894-0347-99-00307-0
- A. S. Fokas, A. R. It⋅s, and A. V. Kitaev, The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395–430. MR 1174420
- Spyridon Kamvissis, Kenneth D. T.-R. McLaughlin, and Peter D. Miller, Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation, Annals of Mathematics Studies, vol. 154, Princeton University Press, Princeton, NJ, 2003. MR 1999840, DOI 10.1515/9781400837182
- S. N. Bernstein, Sur les polynômes orthogonaux relatifs à un segment fini, I: J. Math. Pures Appl. 9 (1930), 127–177; II: J. Math. Pures Appl. 10 (1931), 219–286.
- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
- J. Nuttall and S. R. Singh, Orthogonal polynomials and Padé approximants associated with a system of arcs, J. Approximation Theory 21 (1977), no. 1, 1–42. MR 487173, DOI 10.1016/0021-9045(77)90117-4
- J. Nuttall, Asymptotics of diagonal Hermite-Padé polynomials, J. Approx. Theory 42 (1984), no. 4, 299–386. MR 769985, DOI 10.1016/0021-9045(84)90036-4
- J. Nuttall, Padé polynomial asymptotics from a singular integral equation, Constr. Approx. 6 (1990), no. 2, 157–166. MR 1036606, DOI 10.1007/BF01889355
- S. P. Suetin, On the uniform convergence of diagonal Padé approximants for hyperelliptic functions, Mat. Sb. 191 (2000), no. 9, 81–114 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 9-10, 1339–1373. MR 1805599, DOI 10.1070/SM2000v191n09ABEH000508
- Herbert Stahl, Orthogonal polynomials with complex-valued weight function. I, II, Constr. Approx. 2 (1986), no. 3, 225–240, 241–251. MR 891973, DOI 10.1007/BF01893429
- A. R. It⋅s, A. V. Kitaev, and A. S. Fokas, An isomonodromy approach to the theory of two-dimensional quantum gravity, Uspekhi Mat. Nauk 45 (1990), no. 6(276), 135–136 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 6, 155–157. MR 1101341, DOI 10.1070/RM1990v045n06ABEH002699
- P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
- P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights, Internat. Math. Res. Notices 16 (1997), 759–782. MR 1472344, DOI 10.1155/S1073792897000500
- P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), no. 12, 1491–1552. MR 1711036, DOI 10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for orthogonal polynomials, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 491–501. MR 1648182
- P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
- A. I. Aptekarev and 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 (2004), no. 2, 129–166. MR 2078646, DOI 10.1016/j.jat.2004.06.001
- N. M. Ercolani and K. D. T.-R. McLaughlin, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration, Int. Math. Res. Not. 14 (2003), 755–820. MR 1953782, DOI 10.1155/S1073792803211089
- A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$, Adv. Math. 188 (2004), no. 2, 337–398. MR 2087231, DOI 10.1016/j.aim.2003.08.015
- R. F. Khabibullin, An asymptotic series for Bessel polynomials, Mat. Zametki 77 (2005), no. 6, 948–950 (Russian); English transl., Math. Notes 77 (2005), no. 5-6, 878–881. MR 2246973, DOI 10.1007/s11006-005-0091-2
- Arno B. J. Kuijlaars and Kenneth T.-R. McLaughlin, Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 205–233. MR 1931612, DOI 10.1007/BF03320986
- F. D. Gahov, Kraevye zadachi, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0104117
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
Bibliographic Information
- A. Aptekarev
- Affiliation: The M. V. Keldysh Applied Mathematics Institute, Russian Academy of Sciences, Miusskaya Sq. 4, Moscow 125047, Russia
- MR Author ID: 192572
- Email: aptekaa@keldysh.ru
- R. Khabibullin
- Affiliation: The M. V. Keldysh Applied Mathematics Institute, Russian Academy of Sciences, Miusskaya Sq. 4, Moscow 125047, Russia
- Published electronically: November 15, 2007
- Additional Notes: This work has been supported by the Russian Fund for Fundamental Research (grant No. 05–01–00522), the Support of Leading Scientific Institutions in the RF Programme (grant No. NSh-1551.2003.1), The Mathematical Sciences Department of the Russian Academy of Sciences (programme no. 1) and the INTAS fund (grant No. 03-516637).
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2007, 1-37
- MSC (2000): Primary 42C05
- DOI: https://doi.org/10.1090/S0077-1554-07-00167-7
- MathSciNet review: 2429265