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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.

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