Mathematics of Computation

Author(s): Thomas Ransford; Jérémie Rostand.
Journal: Math. Comp. 76 (2007), 1499-1520.
MSC (2000): Primary 65E05; Secondary 31A15, 90C05
Posted: January 24, 2007
Retrieve article in: PDF DVI PostScript

Abstract: This article introduces a method for computing upper and lower bounds for the logarithmic capacity of a compact plane set. If the set has the Hölder continuity property, then these bounds converge to the value of the capacity. A number of examples are discussed in detail, including the Cantor middle-third set, for which we estimate $c(E)\approx0.220949102189507$ .

1.: L. Baribeau, D. Brunet, T. Ransford, J. Rostand, Iterated function systems, capacity and Green's functions, Comput. Methods Funct. Theory 4 (2004), 47-58. MR 2081665 (2005f:31003)
2.: J. Berntsen, T. O. Espelid, A. Genz, Algorithm 698: DCUHRE: an adaptive multidimensional integration routine for a vector of integrals, ACM Trans. Math. Software 17 (1991), 452-456. MR 1140035
3.: D. G. Cantor, On an extension of the definition of transfinite diameter and some applications, J. Reine Angew. Math. 316 (1980), 160-207. MR 581330 (81m:12002)
4.: L. Carleson, T. W. Gamelin, Complex dynamics, Springer, New York, 1993. MR 1230383 (94h:30033)
5.: L. Carleson, V. Totik, Hölder continuity of Green's functions, Acta Sci. Math. (Szeged) 70 (2004), 557-608. MR 2107529 (2005h:30052)
6.: T. A. Driscoll, Algorithm 756: A MATLAB toolbox for Schwarz-Christoffel mapping, ACM Trans. Math. Software, 22 (1996), 168-186; http://www.math.udel.edu/ driscoll/SC.
7.: M. Embree, L. N. Trefethen, Green's functions for multiply connected domains via conformal mapping, SIAM Rev. 41 (1999), 745-761. MR 1722999 (2000h:30053)
8.: K. J. Falconer, Fractal geometry. Mathematical foundations and applications, 2nd ed., Wiley, Chichester, 2003. MR 2118797 (2006b:28001)
9.: S. Mehrotra, On the Implementation of a Primal-Dual Interior Point Method SIAM J. Optimization 2 (1992), 575-601. MR 1186163 (93g:90047)
10.: Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
11.: Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979), 192-199. MR 534933 (80j:30073)
12.: Ch. Pommerenke, Boundary behaviour of conformal maps, Springer, Berlin, 1992. MR 1217706 (95b:30008)
13.: T. Ransford, Potential theory in the complex plane, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
14.: J. Rostand, Computing logarithmic capacity with linear programming, Experiment. Math. 6 (1997), 221-238 MR 1481591 (98f:65032)
15.: J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, American Mathematical Society, Providence RI, 1969. MR 0218588 (36:1672b)
16.: J. Wang, The theory of games, Oxford University Press, Oxford, 1988. MR 0969605 (90a:90219)
17.: H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math 3 (1969), 127-232. MR 0239059 (39:418)

Retrieve articles in Mathematics of Computation with MSC (2000): 65E05, 31A15, 90C05

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4
Email: ransford@mat.ulaval.ca

Jérémie Rostand
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4
Email: jrostand@mat.ulaval.ca

DOI: 10.1090/S0025-5718-07-01941-2
PII: S 0025-5718(07)01941-2
Keywords: Capacity, matrix game, H\"older continuity property, Cantor set
Received by editor(s): January 18, 2005
Received by editor(s) in revised form: July 6, 2005
Posted: January 24, 2007
Additional Notes: The first author was supported by grants from NSERC and the Canada Research Chairs program
The second author was supported by a grant from NSERC
Copyright of article: Copyright 2007, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS