Abstract
For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE⊂R.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Gonchar, E. A. Rakhmanov (1986):Equilibrium measure and the distribution of zeros of extremal polynomials. Math. USSR-Sb.53:119–130. (Russian original: (1984): Mat. Sb.125(167).)
E. Hille (1982): Analytic Function Theory II. Boston: Ginn.
N. S. Landkof (1972): Foundations of Modern Potential Theory. Berlin: Springer-Verlag.
G. G. Lorentz (1977):Approximation by incomplete polynomials (problems and results). In: Padé and Rational Approximations: Theory and Applications (E. B. Saff, R. S. Varga, eds.). New York: Academic Press, pp. 289–302.
D. S. Lubinsky, H. N. Mhaskar, E. B. Saff (1988):A proof of Freud's conjecture for exponential weights. Constr. Approx.,4:65–83.
D. S. Lubinsky E. B. Saff (1988): Strong Asymptotics for Extremal Polynomials Associated with Weights onR Lecture Notes in Mathematics, vol. 1305. Berlin: Springer-Verlag.
L. S. Luo, J. Nuttall (1987):Asymptotic behavior of the Christoffel function related to a certain unbounded set. In: Approximation Theory, Tampa (E. B. Saff, ed.). Lecture Notes in Mathematics, vol. 1287. Berlin: Springer-Verlag, pp. 105–116.
H. N. Mhaskar (1986):A weighted transfinite diameter. In: Approximation Theory V (C. K. Chui, J. D. Ward, L. L. Schumaker, eds.). New York: Academic Press.
H. N. Mhaskar, E. B. Saff (1984):Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc.,285:204–234.
H. N. Mhaskar, E. B. Saff (1985):Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials.) Constr. Approx.,1:71–91.
E. A. Rakhmanov (1984):On asymptotic properties of polynomials orthogonal on the real axis. Math. USSR.-Sb.47:155–193.
W. Rudin (1974): Real and Complex Analysis. New York: McGraw-Hill.
J. Siciak (1964):Some applications of the method of extremal points. Colloq. Math.,XI:209–249.
H. Stahl (1976): Beitrage zum Problems der Konvergenz von Padé approximierenden. Dissertation, Technischen Universität, Berlin.
H. Stahl (1987):A note on a theorem of H. N. Mhaskar and E. B. Saff. In: Approximation Theory, Tampa (E. B. Saff, ed.). Lecture Notes in Mathematics, vol. 1287. Berlin: Springer-Verlag, pp. 176–179.
H. Stahl (to appear):The convergence of Padé approximants to functions with branch points. J. Approx. Theory.
M. Tsuji (1959): Potential Theory in Modern Function Theory. New York: Chelsea.
H. Widom (1969).Extremal polynomials associated with a system of curves. Adv. in Math.,3:127–232.
Author information
Authors and Affiliations
Additional information
Communicated by Vilmos Totik.
Rights and permissions
About this article
Cite this article
Mhaskar, H.N., Saff, E.B. Weighted analogues of capacity, transfinite diameter, and Chebyshev constant. Constr. Approx 8, 105–124 (1992). https://doi.org/10.1007/BF01208909
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01208909