## Acta Mathematica

### Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials

#### Abstract

Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, ${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$, ${\# A< \infty}$. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function ${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$. The complete proof of Nuttall’s conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

#### Article information

Source
Acta Math., Volume 215, Number 2 (2015), 217-280.

Dates
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.acta/1485802454

Digital Object Identifier
doi:10.1007/s11511-016-0133-5

Mathematical Reviews number (MathSciNet)
MR3455234

Zentralblatt MATH identifier
0863.94012

Rights

#### Citation

Aptekarev, Alexander I.; Yattselev, Maxim L. Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials. Acta Math. 215 (2015), no. 2, 217--280. doi:10.1007/s11511-016-0133-5. https://projecteuclid.org/euclid.acta/1485802454

#### References

• Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions. Dover, New York, 1968.
• Akhiezer, N. I., Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs, 79. Amer. Math. Soc., Providence, RI, 1990.
• Aptekarev, A. I., Sharp constants for rational approximations of analytic functions. Mat. Sb., 193 (2002), 3–72 (Russian); English translation in Sb. Math., 193 (2002), 1–72.
• Aptekarev, A. I., Analysis of the matrix Riemann–Hilbert problems for the case of higher genus and asymptotics of polynomials orthogonal on a system of intervals. Preprints of Keldysh Institute of Applied Mathematics, Russia Acad. Sci., Moscow, 2008.
• Aptekarev, A. I. & Lysov, V. G., Systems of Markov functions generated by graphs and the asymptotics of their Hermite–Padé approximants. Mat. Sb., 201 (2010), 29–78 (Russian); English translation in Sb. Math., 201 (2010), 183–234.
• Aptekarev A. I., Van Assche W.: Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight. J. Approx. Theory 129, 129–166 (2004)
• Baik J., Deift P., McLaughlin K. T.-R., Miller P., Zhou X.: Optimal tail estimates for directed last passage site percolation with geometric random variables. Adv. Theor. Math. Phys. 5, 1207–1250 (2001)
• Baker, G. A. J & Graves-Morris, P., Padé Approximants. Encyclopedia of Mathematics and its Applications, 59. Cambridge Univ. Press, Cambridge, 1996.
• Baratchart L., Stahl H., Yattselev M.: Weighted extremal domains and best rational approximation. Adv. Math. 229, 357–407 (2012)
• Baratchart L., Yattselev M.: Convergent interpolation to Cauchy integrals over analytic arcs. Found. Comput. Math. 9, 675–715 (2009)
• Baratchart L., Yattselev M.: Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights. Int. Math. Res. Not. 22, 4211–4275 (2010)
• Baratchart L., Yattselev M.: Padé approximants to certain elliptic-type functions. J. Anal. Math. 121, 31–86 (2013)
• Bertola M., Mo M. Y.: Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights. Adv. Math. 220, 154–218 (2009)
• Deift, P., Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, 3. Amer. Math. Soc., Providence, RI, 1999.
• Deift P., Kriecherbauer T., McLaughlin K. D. T.-R., Venakides S., Zhou X.: 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, 1335–1425 (1999)
• Deift P., Kriecherbauer T., McLaughlin K. D. T.-R., Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491–1552 (1999)
• Deift P., Zhou X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. 137, 295–368 (1993)
• Dieudonné, J., Foundations of Modern Analysis. Pure and Applied Mathematics, 10-I. Academic Press, New York–London, 1969.
• Dumas, S., Sur le déveleppement des fonctions elliptiques en fractions continues. Ph.D. Thesis, Universität Zürich, Zürich, 1908.
• Fokas A. S., Its A. R., Kitaev A. V.: Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142, 313–344 (1991)
• Fokas A. S., It.s A. R., Kitaev A. V.: The isomonodromy approach to matrix models in 2D quantum gravity. Comm. Math. Phys. 147, 395–430 (1992)
• Gakhov, F. D., Boundary Value Problems. Dover, New York, 1990.
• Gammel, J. L. & Nuttall, J., Note on generalized Jacobi polynomials, in The Riemann Problem, Complete Integrability and Arithmetic Applications (Bures-sur-Yvette/New York, 1979/1980), Lecture Notes in Math., 925, pp. 258–270. Springer, Berlin–New York, 1982.
• Goluzin, G.M., Geometric Theory of Functions of a Complex Variable. Translations of Mathematical Monographs, 26. Amer. Math. Soc., Providence, RI, 1969.
• Gonchar, A.A., The rate of rational approximation of certain analytic functions. Mat. Sb., 105(147) (1978), 147–163 (Russian); English translation in Math. USSR–Sb., 34 (1978), 164–179.
• Gonchar, A. A. & López Lagomasino, G., Markov’s theorem for multipoint Padé approximants. Mat. Sb., 105(147) (1978), 512–524 (Russian); English translation in Math. USSR–Sb., 34 (1978), 449–459.
• Gonchar, A.A. & Rakhmanov, E. A., Equilibrium distributions and the rate of rational approximation of analytic functions. Mat. Sb., 134(176) (1987), 306–352 (Russian); English translation in Math. USSR–Sb., 62 (1989), 305–348.
• Kamvissis, S., McLaughlin, K. D. T.-R. & Miller, P. D., Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation. Annals of Mathematics Studies, 154. Princeton Univ. Press, Princeton, NJ, 2003.
• Kamvissis, S. & Rakhmanov, E. A., Existence and regularity for an energy maximization problem in two dimensions. J. Math. Phys., 46 (2005), 083505, 24 pp.
• Kriecherbauer T., McLaughlin K. D. T.-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 6, 299–333 (1999)
• Kuijlaars A. B. J., Martínez-Finkelshtein A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math. 94, 195–234 (2004)
• Kuijlaars A. B. J., McLaughlin K. T. R., Van Assche W., Vanlessen M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [-1, 1]. Adv. Math. 188, 337–398 (2004)
• Kuijlaars A. B. J., McLaughlin K. D. T.-R.: Riemann–Hilbert analysis for Laguerre polynomials with large negative parameter. Comput. Methods Funct. Theory 1, 205–233 (2001)
• Kuijlaars A. B. J., McLaughlin K. D. T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constr. Approx. 20, 497–523 (2004)
• Martínez-Finkelshtein A., Rakhmanov E. A.: Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials. Comm. Math. Phys. 302, 53–111 (2011)
• Martínez-Finkelshtein, A., Rakhmanov, E. A. & Suetin, S.P., Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later, in Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Contemp. Math., 578, pp. 165–193. Amer. Math. Soc., Providence, RI, 2012.
• Nikishin E. M.: On the convergence of diagonal Padé approximants to certain functions. Math. USSR Sb. 30, 249–260 (1976)
• Nikishin, E. M. & Sorokin, V. N., Rational Approximations and Orthogonality. Translations of Mathematical Monographs, 92. Amer. Math. Soc., Providence, RI, 1991.
• Nuttall, J., The convergence of Padé approximants to functions with branch points, in Padé and Rational Approximation (Tampa, FL, 1976), pp. 101–109. Academic Press, New York, 1977.
• Nuttall, J., Sets of minimum capacity, Padé approximants and the bubble problem, in Bifurcation Phenomena in Mathematical Physics and Related Topics (Dordrecht, 1980), pp. 185–201. Reidel, 1980.
• Nuttall J.: Asymptotics of diagonal Hermite–Padé polynomials. J. Approx. Theory 42, 299–386 (1984)
• Nuttall J.: Asymptotics of generalized Jacobi polynomials. Constr. Approx. 2, 59–77 (1986)
• Nuttall J.: Padé polynomial asymptotics from a singular integral equation. Constr. Approx. 6, 157–166 (1990)
• Nuttall J., Singh S. R.: Orthogonal polynomials and Padé approximants associated with a system of arcs. J. Approx. Theory 21, 1–42 (1977)
• Padé H.: Sur la représentation approchée d’une fonction par des fractions rationnelles. Ann. Sci. ´ Ecole Norm. Sup. 9, 3–93 (1892)
• Perevoznikova, E. A. & Rakhmanov, E. A., Variation of the equilibrium energy and S-property of compacta of minimal capacity. Manuscript, 1994.
• Pommerenke, C., Univalent Functions. Studia Mathematica/Mathematische Lehrbücher, 25. Vandenhoeck & Ruprecht, Göttingen, 1975.
• Ransford, T., Potential Theory in the Complex Plane. London Mathematical Society Student Texts, 28. Cambridge Univ. Press, Cambridge, 1995.
• Saff, E. B. & Totik, V., Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, 316. Springer, Berlin–Heidelberg, 1997.
• Stahl, H., Extremal domains associated with an analytic function. I, II. Complex Variables Theory Appl., 4 (1985), 311–324, 325–338.
• Stahl H.: The structure of extremal domains associated with an analytic function. Complex Variables Theory Appl. 4, 339–354 (1985)
• Stahl, H., Orthogonal polynomials with complex-valued weight function. I, II. Constr. Approx., 2 (1986), 225–240, 241–251.
• Stahl H.: On the convergence of generalized Padé approximants. Constr. Approx. 5, 221–240 (1989)
• Stahl, H., Diagonal Padé approximants to hyperelliptic functions. Ann. Fac. Sci. Toulouse Math., Special issue (1996), 121–193.
• Stahl H.: The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91, 139–204 (1997)
• Suetin, S.P., On the uniform convergence of diagonal Padé approximants for hyperelliptic functions. Mat. Sb., 191 (2000), 81–114 (Russian); English translation in Sb. Math., 191 (2000), 1339–1373.
• Suetin, S.P., On the convergence of Chebyshev continued fractions for elliptic functions. Mat. Sb., 194 (2003), 63–92 (Russian); English translation in Sb. Math., 194 (2003), 1807–1835.
• Szegő, G., Orthogonal Polynomials. Colloquium Publications, 23. Amer. Math. Soc., Providence, RI, 1975.