Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T16:38:02.808Z Has data issue: false hasContentIssue false
  • English
  • Français

On convergence in capacity

Published online by Cambridge University Press:  17 April 2009

Burnett Meyer
Burnett Meyer
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado, USA.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The (logarithmic) capacity or transfinite diameter is originally defined for compact sets in the complex plane. An extension may be made by defining the capacity of a given arbitrary set in the plane as the supremum of the capacities of all compact sets contained in the given set. Convergence in capacity is defined analogously to convergence in measure. It is shown in this paper that properties of convergence in capacity are also analogous to those of convergence in measure.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 1 , February 1976 , pp. 1 - 5
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Goluzin, G.M., Geometric theory of functions of a complex variable (Translations of Mathematical Monographs, 26. Amer. Math. Soc., Providence, Rhode Island, 1969).CrossRefGoogle Scholar
[2]Hille, Einar, Analytic function theory, Vol. II (Ginn, Boston, New York, Chicago, Atlanta, Dallas, Palo Alto, Toronto, 1962).Google Scholar
[3]Jones, William B. and Thron, W.J., “On the convergence of Padé approximants”, SIAM J. Math. Anal. 6 (1975), 9–16.CrossRefGoogle Scholar
[4]Nuttall, J., “The convergence of Padé approximants of meromorphic functions”, J. Math. Anal. Appl. 31 (1970), 147–153.CrossRefGoogle Scholar
[5]Pommnerenke, Ch., “Padé approximants and convergence in capacity”, J. Math. Anal. Appl. 41 (1973), 775–780.CrossRefGoogle Scholar
[6]Rudin, Walter, Real and complex analysis (McGraw-Hill, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1966).Google Scholar
[7]Taylor, Angus E., General theory of functions and integration (Blaisdell, New York, Toronto, London, 1965).Google Scholar
[8]Tsuji, M., Potential theory in modern function theory (Maruzen, Tokyo, 1959).Google Scholar