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Orthogonal measures on the boundary of a Riemann surface and polynomial hull of compacts of finite length. (English) Zbl 0951.32005
Summary: Let $$\mu$$ be an orthogonal measure with compact support of finite length in $$\mathbb{C}^n$$. The author proves, under a very weak hypothesis of regularity on the support $$\text{(Supp} \mu)$$ of $$\mu$$, that this measure is characterized by its boundary values (in the weak sense of currents) of the current $$[T]\wedge\varphi$$, where $$T$$ is an analytic subset of dimension 1 of $$\mathbb{C}^n \setminus\text{Supp} \mu$$ and $$\varphi$$ is a holomorphic $$(1,0)$$-form on $$T$$. This allows to prove that the polynomial hull $$\widehat X$$ of a compactum $$X\subset\mathbb{C}^n$$ of finite length with a weak regularity assumption is its union with an analytic subset of pure dimension 1 of $$\mathbb{C}^n \setminus X$$. He also proves that the measure $$\mu$$ can be decomposed into a sum of orthogonal measures with small support. He deduces that a continuous function on $$\widehat X$$ is approximable by polynomials if and only if it is locally approximable.

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 32F17 Other notions of convexity in relation to several complex variables
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