Vol. 58, No. 2, 1975

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Cauchy transforms and characteristic functions

Kenneth Clayton Pietz

Vol. 58 (1975), No. 2, 563–568

The following problem arises in the study of rational approximation: classify all plane sets E such that μ(z) (ζ)(ζ z) = χE(z) area almost everywhere for some complex Borel measure μ. A partial solution to this problem for compact sets is given here. The main result is the following.

THEOREM. Let K be a compact plane set with connected dense interior. Then there is a measure μ such that μ = χK area a.e., if and only if K has finite Painlevé length.

Mathematical Subject Classification
Primary: 30A98
Received: 10 June 1974
Published: 1 June 1975
Kenneth Clayton Pietz