Vol. 26, No. 3, 1968

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On a theorem of Mergelyan

John Brady Garnett

Vol. 26 (1968), No. 3, 461–467

The following is a theorem of S. N. Mergelyan.

Theorem. If X is a compact plane set with finitely many complementary components, then every continuous function on X which is analytic on the interior is a uniform limit on X of rational functions.

Here two short proofs of the theorem are given. Both proofs are obtained by considering the measures on X orthogonal to the algebra of rational functions.

Mathematical Subject Classification
Primary: 46.55
Received: 25 May 1967
Published: 1 September 1968
John Brady Garnett
Department of Mathematics
University of California
Los Angeles CA 90024
United States