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Application of half-order differentials on Riemann surfaces to quadrature identities for arc-length. (English) Zbl 0652.30029

A region \(\Omega\) in \({\mathbb{C}}\) is called a quadrature domain for arc- length if there exist finitely many points \(z_ k\in \Omega\) and complex numbers \(a_{kj}\) (1\(\leq k\leq m\), \(0\leq j\leq n_ k-1)\) such that \[ \int_{\partial \Omega}f(z)| dz| =\sum^{m}_{k=1}\sum^{n_ k-1}_{j=0}a_{kj}f^{(j\quad)}(z_ k) \] for every function f in some suitable test class of holomorphic functions on \(\Omega\). Here \(\partial \Omega\) is assumed to consist of finitely many components each of which is a continuum of finite one-dimensional Hausdorff measure. In this paper the test class is typically the Hilbert space E 2(\(\Omega)\). The analogous situation of quadrature domains for area measure has been investigated earlier by a number of authors.
The main result of the paper is a characterization of quadrature domains for arc length in terms of conformal mappings from standard regions. Suppose W is a finitely connected region in \({\mathbb{C}}\) such that each component of \(\partial W\) is an analytic closed Jordan curve. Let \(\hat W\) denote the Schottky double of W across \(\partial W\). Assume g:W\(\to \Omega\) is a conformal mapping. Then \(\Omega\) is a quadrature domain for arc-length if and only if \(\sqrt{dg}\) extends to a meromorphic half-order differential on \(\hat W.\) In addition, the existence of quadrature domains for arc-length of all conformal types under consideration is established by showing that for a given W there always exist univalent functions g on W such that \(\sqrt{dg}\) extends to a meromorphic half- order differential on \(\hat W.\) The author also investigates quadrature domains containing \(\infty\) and gives other characterizations of quadrature domains.
Reviewer: D.Minda

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30D55 \(H^p\)-classes (MSC2000)
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