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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.

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) |

Full Text:
DOI

### References:

[1] | D. Aharonov and H. S. Shapiro,Domains on which analytic functions satisfy quadrature identities, J. Analyse Math.30 (1976), 39–73. · Zbl 0337.30029 |

[2] | C. Auderset,Sur le théorème d’approximation de Runge, Enseign. Math.26 (1980), 219–224. · Zbl 0475.30026 |

[3] | Y. Avci,Quadrature identities and the Schwarz function, Doctoral Dissertation, Stanford University, 1977. |

[4] | Pl. J. Davis,The Schwarz Function and its Applications, The Carus Mathematical Monographs 17, The Mathematical Association of America, 1974. |

[5] | P. Duren,Theory of H p Spaces, Academic Press, New York, 1970. · Zbl 0215.20203 |

[6] | R. C. Gunning,Lectures on Riemann Surfaces, Princeton University Press, Princeton, 1966. · Zbl 0175.36801 |

[7] | B. Gustafsson,Quadrature identities and the Schottky double, Acta Appl. Math.1 (1983), 209–240. · Zbl 0559.30039 |

[8] | D. Hejhal,Theta Functions, Kernel Functions, and Abelian Integrals, Memoirs Am. Math. Soc. no. 129, Stanford, 1972. |

[9] | C. Pommerenke,Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. |

[10] | W. Rudin,Analytic functions of class H p, Trans. Am. Math. Soc.78 (1955), 46–66. · Zbl 0064.31203 |

[11] | M. Schiffer,Half-order differentials on Riemann surfaces, SIAM J. Appl. Math.14 (1966), 922–934. · Zbl 0164.37903 |

[12] | M. Schiffer and N. S. Hawley,Half-order differentials on Riemann surfaces, Acta Math.115 (1966), 199–236. · Zbl 0136.06701 |

[13] | H. S. Shapiro,Domains allowing exact quadrature identities for harmonic functions–an approach based on PDE, inAnniversary Volume on Approximation Theory and Functional Analysis, P. L. Butzer, R. L. Stens and B. Sz.-Nagy (ed.), ISNM 65, Birkhäuser-Verlag, Basel, Boston, Stuttgart, 1984. |

[14] | H. S. Shapiro and C. Ullemar,Conformal mappings satisfying certain extremal properties, and associated quadrature identities, Royal Institute of Technology research report TRITA-MAT-1986-6, Stockholm, 1981. |

[15] | G. Toumarkine and S. Havinson,On the definition of analytic functions of class E p in multiply connected domains, Uspehi Mat. Nauk13 (1958), 201–206 (in Russian). |

[16] | G. Toumarkine and S. Havinson,On the decomposition theorem for analytic functions of class E p in multiply connected domains, Uspehi. Mat. Nauk13 (1958), 223–228 (in Russian). · Zbl 0093.27401 |

[17] | G. Toumarkine and S. Havinson,Classes de fonctions analytiques dans des domains multiplement connexes, inFonctions d’une variables complexe, Problèmes contemporains, A. I. Markouchevitch (red.), Gauthiers-Villars, Paris, 1962, pp. 37–71. |

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