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On Uniform Approximation by Rational Functions with an Application to Chordal Cluster Sets*

Published online by Cambridge University Press:  22 January 2016

J.T. Gresser
J.T. Gresser*
Affiliation:
University of Wisconsin-Milwaukee
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For a closed and bounded set E in the complex plane, let A(E) denote the collection of all functions continuous on E and analytic on E°, its interior; let R(E) denote the collection of all functions which are uniform limits on E of rational functions with poles outside E. Then let A denote the collection of all closed, bounded sets for which A(E) = R(E). The purpose of this paper is to formulate a condition on a set, which is essentially of a geometric nature, in order that the set belong to A. Then using approximation techniques, we shall construct a meromorphic function having a certain boundary behavior on a perfect set; this answers a question raised in [1].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 34 , March 1969 , pp. 143 - 148
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

Footnotes

*

The results presented here are part of the author’s doctoral dissertation written at the University of Wisconsin-Milwaukee under the direction of Professor F. Bagemihl.

References

[1] Bagemihl, F., Piranian, G. and Young, G.S., Intersections of cluster sets, Bui. Inst. Politehn. Ia§i (N.S.) 5 (1959), 29–34.Google Scholar
[2] Bagemihl, F., Some results and problems concerning chordal principal cluster sets, Nagoya Math. J. 29 (1967), 7–18.CrossRefGoogle Scholar
[3] Collingwood, E.F. and Lohwater, A.J., The theory of cluster sets, Cambridge, 1966.CrossRefGoogle Scholar
[4] Noshiro, K., Cluster sets, Berlin, 1960.CrossRefGoogle Scholar
[5] Walsh, J.L., Interpolation and approximation by rational functions in the complex domain. 3rd ed., Providence, 1960.Google Scholar
[6] Vituskin, A.G., Necessary and sufficient conditions on a set in order that any continuous function analytic at the interior points of the set may admit of uniform approximation by rational functions, Dokl. Akad. Nauk SSSR 171 (1966), 1255–1258= Soviet Math. Dokl. 7 (1966), 1622–1625.Google Scholar