Strong asymptotics of polynomials orthogonal with respect to

S. P. Suetin

Sbornik. Mathematics, Tome 200 (2009) no. 1, pp. 77-93

Résumé

For polynomials orthogonal with respect to a complex-valued weight on the closed interval $\Delta=[-1,1]$ a strong asymptotic formula in a neighbourhood of $\Delta$ is obtained. In particular, for the ‘trigonometric’ weight $\rho_0(x)=e^{ix}$, $x\in\Delta$, this formula yields a description of the asymptotic behaviour of each of the $n$ zeros of the $n$th orthogonal polynomial as $n\to\infty$. This strong asymptotic formula is deduced on the basis of Nuttall's singular integral equation. Bibliography: 28 titles.

Détail

Keywords: strong asymptotics.
Mots-clés : Padé approximants, orthogonal polynomials

Bibliographie

[1] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, RI, 1959 | MR | Zbl | Zbl

[2] N. I. Akhiezer, “Function theory according to Chebyshev”, Mathematics of the 19th century, Birkhäuser, Basel, 1998, 1–81 | MR | MR | Zbl | Zbl

[3] P. L. Chebyshev, “O nepreryvnykh drobyakh”, Uchenye zapiski Imp. Akademii Nauk, III (1855), 636–664; P. Tchébycheff, “Sur les fractions continues”, Journ. de Math. Pures et Appl. Sér. 2, 3 (1858), 289–323; “О непрерывных дробях”, Полное собрание сочинений, т. II, Изд-во АН СССР, М., Л., 1947, 103–126 | MR

[4] A. A. Markov, “O raspredelenii kornei nekotorykh uravnenii”, Soobsch. i protokoly zased. matem. o-va pri Kharkovskom un-te, ser. 2, no. 2, 1886, 89–98; Избранные труды по теории непрерывных дробей и теории функций, наименее уклоняющихся от нуля, Гостехиздат, М.–Л., 1948, 33–43 | MR | Zbl

[5] S. Dumas, Sur le développement des fonctions elliptiques en fractions continues, Thèse, Zürich, 1908 | Zbl

[6] S. N. Bernshtein, O mnogochlenakh, ortogonalnykh v konechnom intervale, ONTI, Kharkov, 1937; “О многочленах, ортогональных на конечном отрезке”, Собрание сочинений. Т. 2. Конструктивная теория функций [1931–1953], Изд-во АН СССР, М., 1954, 7–79 | MR | Zbl

[7] Ya. L. Geronimus, Teoriya ortogonalnykh mnogochlenov, Obzor dostizhenii otechestvennoi matematiki, GITTL, M.–L., 1950 | MR

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

[9] A. A. Gonchar, S. P. Suetin, Ob approksimatsiyakh Pade meromorfnykh funktsii markovskogo tipa, Sovr. probl. matem., 5, MIAN, M., 2004 | DOI | MR | Zbl

[10] 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

[11] A. A. Gončar, “On convergence of Padé approximants for some classes of meromorphic functions”, Math. USSR-Sb., 26:4 (1975), 555–575 | DOI | MR | Zbl

[12] A. P. Magnus, “Toeplitz matrix techniques and convergence of complex weight Padé approximants”, J. Comput. Appl. Math., 19:1 (1987), 23–38 | DOI | MR | Zbl

[13] J. Nuttall, “Asymptotics of diagonal Hermite–Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[14] A. I. Aptekarev, “Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices”, Math. USSR-Sb., 53:1 (1986), 233–260 | DOI | MR | Zbl

[15] H. Stahl, “Orthogonal polynomials with complex-valued weight function, I”, Constr. Approx., 2:1 (1986), 225–240 | DOI | MR | Zbl

[16] H. Stahl, “Orthogonal polynomials with complex-valued weight function, II”, Constr. Approx., 2:1 (1986), 241–251 | DOI | MR | Zbl

[17] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Math. USSR-Sb., 62:2 (1989), 305–348 | DOI | MR | Zbl | Zbl

[18] A. L. Lukashov, “Obobschenie klassicheskikh ortogonalnykh mnogochlenov na sluchai dvukh promezhutkov”, Fundament. i prikl. matem., 5:4 (1999), 1103–1110 | MR | Zbl

[19] A. L. Lukashov, “Circular parameters of polynomials orthogonal on several arcs of the unit circle”, Sb. Math., 195:11 (2004), 1639–1663 | DOI | MR | Zbl

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

[21] L. A. Knizhnerman, “The Gauss–Arnoldi quadrature for the function $\langle(zI-A)^{-1}\varphi, \varphi\rangle$ and Padé-like rational approximation of Markov-type functions”, Sb. Math., 199:2 (2008), 185–206 | DOI | MR

[22] A. A. Gonchar, “Singular points of meromorphic functions defined by their expansion in a $C$-fraction”, Sb. Math., 197:10 (2006), 1405–1416 | DOI | MR | Zbl

[23] A. P. Starovoitov, N. A. Starovoitova, “Padé approximants of the Mittag–Leffler functions”, Sb. Math., 198:7 (2007), 1011–1023 | DOI | MR | Zbl

[24] S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141 | DOI | MR | Zbl

[25] S. P. Suetin, “Trace formulae for a class of Jacobi operators”, Sb. Math., 198:6 (2007), 857–885 | DOI | MR | Zbl

[26] S. P. Suetin, “Strong asymptotics of zeros of polynomials orthogonal with respect to a complex-valued weight”, Russian Math. Surveys, 62:4 (2007), 823–825 | DOI | MR | Zbl

[27] 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 | Zbl

[28] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher, 184, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | MR | Zbl