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On mother body measures with algebraic Cauchy transform. (English) Zbl 1364.31003

The paper deals with the existence of mother body measures with algebraic Cauchy transform. More precisely, the authors consider the following question: given a germ \(f\) of an algebraic function near \(\infty\), is it possible to find a compactly supported signed measure whose Cauchy transform coincides with the analytic continuation of \(f\) in \(\mathbb{C}\)? Moreover, for which \(f\) is such a measure positive? The authors provide several results on some specific classes of algebraic functions, formulate conjectures, and pose open problems.

MSC:

31A25 Boundary value and inverse problems for harmonic functions in two dimensions
35R30 Inverse problems for PDEs
86A22 Inverse problems in geophysics
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