Abstract
Letf be an analytic function with all its singularities in a compact set\(E_f \subset \bar C\) of (logarithmic) capacity zero. The function may have branch points. The convergence of generalized (multipoint) Padé approximants to this type of function is investigated. For appropriately selected schemes of interpolation points, it is shown that close-to-diagonal sequences of generalized Padé approximants converge in capacity tof in a certain domain that can be characterized by the property of the minimal condenser capacity. Using a pole elimination procedure, another set of rational approximants tof is derived from the considered generalized Padé approximants. These new approximants converge uniformly on a given continuum\(V \subset \bar C\backslash E_f\) with a rate of convergence that has been conjectured to be best possible. The continuumV is assumed not to divide the complex plane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Bagby (1967):The modulus of a plane condenser. J. Math. Mech.,17:315–329.
G. A. Baker, Jr.,P. R. Graves-Morris (1981): Padé Approximants, Parts I and II. Encyclopedia of Mathematics and Its Applications, vols. 13 and 14. Reading, MA: Addison Wesley.
A. A. Gonchar (1975):On the convergence of generalized Padé approximants of meromorphic functions. Math. USSR-sb.,27:503–514.
A. A. Gonchar (1982):Rational approximation of analytic functions. In: Linear and Complex Analysis Problem Book (V. P. Havin, S. V. Hruscev, N. K. Nikolskii, eds.). Lecture Notes in Mathematics, vol. 1043. Berlin: Springer-Verlag, pp. 471–474.
A. A. Gonchar, G. Lopes (1978):On Markov's theorem for multipoint Padé approximants. Math. USSR-sb.,34:449–459.
W. K. Hayman (1958): Multivalent Functions. Cambridge: Cambridge University Press.
N. S. Landkof (1972): Foundations of Modern Potential Theory. Berlin: Springer-Verlag.
D. S. Lubinsky (1983):Divergence of complex rational approximations. Pacific J. Math.,108:141–153.
B. Meyer (1976):On convergence in capacity. Bull. Austral. Math. Soc.,14:1–5.
R. Nevanlinna (1953): Eindeutige analytische Funktionen. Berlin: Springer-Verlag.
J. Nuttall (1970):The convergence of Padé approximants of meromorphic functions. J. Math. Anal. Appl.,31:147–153.
J. Nuttall (1977):The convergence of Padé approximants to functions with branch points. In: Padé and Rational Approximation (E. B. Saff, R. S. Varga, eds.). New York: Academic Press, pp. 101–109.
J. Nuttall (1982): The convergence of Padé Approximants and Their Generalizations. Lecture Notes in Mathematics, vol. 925, Berlin: Springer-Verlag, pp. 246–257.
J. Nuttall (1985): On Sets of Minimal Capacity. Lecture Notes in Pure and Applied Mathematics, New York: Marcel Dekker.
CH. Pommerenke (1973):Padé approximants and convergence in capacity. J. Math. Anal. Appl.,41:775–780.
E. A. Rahmanov (1980):On the convergence of Padé approximants in classes of holomorphic functions. Math. USSR-sb.,40:149–155.
E. B. Saff (1972):An extension of Montessus de Ballore's theorem on convergence of interpolating rational functions. J. Approx. Theory,6:63–67.
H.Stahl (1976): Beiträge zum Problem der Konvergenz von Padé Approximierenden. Doctoral dissertation, TU-Berlin. (A summary is contained in H. Stahl (1986):Non-diagonal Padé approximants to Markov functions. In: Approximation Theory, V (C. K. Chui, L. L. Schumaker, J, D. Ward, eds.). Orlando; Academic Press, pp. 571–574.)
H. Stahl (1985):Extremal domains associated with an analytic function, I, II. Complex Variables Theory Appl.,4:311–338.
H. Stahl (1985): On the Divergence of Certain Padé Approximants and the Behaviour of the Associated Orthogonal Polynomials. Lecture Notes in Mathematics, vol. 1171. Berlin: Springer-Verlag, pp. 321–330.
H. Stahl (1986):Orthogonal polynomials with complex-valued weight function, I. Constr. Approx.,2:225–240.
H. Stahl (1986):Orthogonal polynomials with complex-valued weight function, II. Constr. Approx.,2:241–251.
H. Stahl (1986):Existence and uniqueness of rational interpolants with free and prescribed poles. Lecture Notes in Mathematics, vol. 1287. Berlin: Springer-Verlag, pp. 180–208.
H.Stahl (submitted):The convergence of Padé approximants to functions with branch points. J. Approx. Theory.
H. Wallin (1979): Potential Theory and Approximation of Analytic Functions by Rational Interpolation. Lecture Notes in Mathematics, vol. 797. Berlin: Springer-Verlag, pp. 434–450.
Author information
Authors and Affiliations
Additional information
Communicated by Dieter Gaier.
Rights and permissions
About this article
Cite this article
Stahl, H. On the convergence of generalized Padé approximants. Constr. Approx 5, 221–240 (1989). https://doi.org/10.1007/BF01889608
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01889608