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September 2003 The Capacity of some Sets in the Complex Plane
Maurice Hasson
Bull. Belg. Math. Soc. Simon Stevin 10(3): 421-436 (September 2003). DOI: 10.36045/bbms/1063372347

Abstract

For a positive integer $s$ and for $0\le a< b,$ let $$K=K^s_{a,b}=\bigcup_{k=0}^{s-1}e^{2\pi i\frac{k}{s}}[a,b].$$ We find that the {\it capacity} $\mathrm{Cap} (K)$ of $K$ is $$\mathrm{Cap} (K) = \sqrt [s]{\frac{b^s-a^s}4}\cdot\eqno (1)$$ \par From this relation we derive several classical results, due to Akhiezer, Henrici, and Bartolomeo and He, on capacities of some sets in the complex plane. \par An extension relation (1) to more general sets in the complex plane, together with potential theoretic techniques, is then used to obtain {\it saturation} theorems pertaining to approximation by polynomials with integer coefficients.

Citation

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Maurice Hasson. "The Capacity of some Sets in the Complex Plane." Bull. Belg. Math. Soc. Simon Stevin 10 (3) 421 - 436, September 2003. https://doi.org/10.36045/bbms/1063372347

Information

Published: September 2003
First available in Project Euclid: 12 September 2003

zbMATH: 1037.31001
MathSciNet: MR2017453
Digital Object Identifier: 10.36045/bbms/1063372347

Subjects:
Primary: 31E10, 41A10, 41A29, 41A40

Rights: Copyright © 2003 The Belgian Mathematical Society

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Vol.10 • No. 3 • September 2003
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