Cauchy transforms and characteristic functions

Kenneth Clayton Pietz

Vol. 58 (1975), No. 2, 563–568

DOI: 10.2140/pjm.1975.58.563

Abstract

The following problem arises in the study of rational approximation: classify all plane sets E such that μ(z) ≡∫ dμ(ζ)∕(ζ − z) = χ_E(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 = χ_K area a.e., if and only if K has finite Painlevé length.

Mathematical Subject Classification

Primary: 30A98

Milestones

Received: 10 June 1974

Published: 1 June 1975