Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight

N. R. Ikonomov ; R. K. Kovacheva ; S. P. Suetin

Izvestiya. Mathematics, Tome 79 (2015) no. 6, pp. 1215-1234

Résumé

We obtain Nuttall's integral equation provided that the corresponding complex-valued function $\sigma(x)$ does not vanish and belongs to the Dini–Lipschitz class. Using this equation, we obtain a complex analogue of Bernshtein's classical asymptotic formulae for polynomials orthogonal on the closed unit interval $\Delta=[-1,1]$ with respect to a complex-valued weight $h(x)=\sigma(x)/\sqrt{1-x^2}$.

Détail

Keywords: strong asymptotics, Bernshtein's formula, Nuttall's method.
Mots-clés : orthogonal polynomials, Padé polynomials

Bibliographie

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

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

[3] A. I. Aptekarev, M. L. Yattselev, Pade approximants for functions with branch points – strong asymptotics of Nuttall–Stahl polynomials, 2011, 45 pp., arXiv: 1109.0332

[4] M. L. Yattselev, “Nuttall's theorem with analytic weights on algebraic $S$-contours”, J. Approx. Theory, 190 (2015), 73–90 | DOI | MR | Zbl

[5] J. Nuttall, G. M. Trojan, “Asymptotics of Hermite–Padé polynomials for a set of functions with different branch points”, Constr. Approx., 3:1 (1987), 13–29 | DOI | MR | Zbl

[6] A. I. Aptekarev, A. B. J. Kuijlaars, W. Van Assche, “Asymptotics of Hermite–Padé rational approximants for two analytic functions with separated pairs of branch points (case of genus $0$)”, Int. Math. Res. Pap. IMRP, 2008:4 (2008), rpm007, 128 pp. | DOI | MR | Zbl

[7] 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

[8] A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight. II”, Sb. Math., 205:9 (2014), 1334–1356 | DOI | DOI | MR | Zbl

[9] S. P. Suetin, “Comparative asymptotics 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

[10] A. A. Gonchar, S. P. Suetin, “On Padé approximants of Markov-type meromorphic functions”, Proc. Steklov Inst. Math., 272: suppl. 2 (2011), S58–S95 | DOI | DOI | MR | Zbl

[11] A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131 | DOI | DOI | MR | Zbl

[12] S. N. Bernshtein, O mnogochlenakh, ortogonalnykh v konechnom intervale, ONTI, Kharkov, 1937, 128 pp.

[13] S. P. Suetin, “Strong asymptotics of polynomials orthogonal with respect to a complex weight”, Sb. Math., 200:1 (2009), 77–93 | DOI | DOI | MR | Zbl

[14] 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

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

[16] J. Nuttall, R. S. Singh, “Orthogonal polynomials and Padé approximants associated with a system of arcs”, J. Approximation Theory, 21:1 (1977), 1–42 | DOI | MR | Zbl

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

[18] N. I. Akhiezer, “Chebyshevskoe napravlenie v teorii funktsii”, Matematika XIX veka, Vyp. 3, eds. A. N. Kolmogorov, A. P. Yushkevich, Nauka, M., 1987, 9–79 | MR | Zbl

[19] P. L. Chebyshev, “O nepreryvnykh drobyakh”, Uchenye zap. Imp. Akad. Nauk, III (1855), 636–664; ПолноРμ собраниРμ сочинРμРЅРёРNo, С‚. II, Р�Р·Рґ-РІРѕ РђРќ РЎРЎРЎР , Рњ.–Р›., 1947, 103–126 ; P. Tchébycheff, “Sur les fractions continues”, J. Math. Pures Appl. (2), 3 (1858), 289–323 | MR | Zbl

[20] W. Stekloff (V. Steklov), “Sur le développement des fonctions continues en séries de polynômes de Tchébychef”, Izvѣstiya Rossiiskoi Akademii Nauk'. VI seriya, 15 (1921), 249–266 | Zbl

[21] H. Stahl, “Three different approaches to a proof of convergence for Padé approximants”, Rational approximation and applications in mathematics and physics (Łańcut, 1985), Lecture Notes in Math., 1237, Springer, Berlin, 1987, 79–124 | DOI | MR | Zbl

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

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

[24] S. Dumas, Sur le développement des fonctions elliptiques en fractions continues, Thèse, Zürich, 1908, 59 pp. | Zbl

[25] A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72 | DOI | DOI | MR | Zbl

[26] 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

[27] E. M. Nikishin, V. N. Sorokin, Rational approximations and orthogonality, Transl. Math. Monogr., 92, Amer. Math. Soc., Providence, RI, 1991, viii+221 pp. | MR | MR | Zbl | Zbl

[28] A. A. Gonchar, “Rational approximation of analytic functions”, Proc. Steklov Inst. Math., 272: suppl. 2 (2011), S44–S57 | DOI | DOI | MR | Zbl

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

[30] A. Zygmund, Trigonometric series, v. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp., vii+354 pp. | MR | MR | Zbl | Zbl