## 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)

### Keywords:

quadrature domain; Schottky double; half-order differential
Full Text:

### References:

  D. Aharonov and H. S. Shapiro,Domains on which analytic functions satisfy quadrature identities, J. Analyse Math.30 (1976), 39–73. · Zbl 0337.30029  C. Auderset,Sur le théorème d’approximation de Runge, Enseign. Math.26 (1980), 219–224. · Zbl 0475.30026  Y. Avci,Quadrature identities and the Schwarz function, Doctoral Dissertation, Stanford University, 1977.  Pl. J. Davis,The Schwarz Function and its Applications, The Carus Mathematical Monographs 17, The Mathematical Association of America, 1974.  P. Duren,Theory of H p Spaces, Academic Press, New York, 1970. · Zbl 0215.20203  R. C. Gunning,Lectures on Riemann Surfaces, Princeton University Press, Princeton, 1966. · Zbl 0175.36801  B. Gustafsson,Quadrature identities and the Schottky double, Acta Appl. Math.1 (1983), 209–240. · Zbl 0559.30039  D. Hejhal,Theta Functions, Kernel Functions, and Abelian Integrals, Memoirs Am. Math. Soc. no. 129, Stanford, 1972.  C. Pommerenke,Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.  W. Rudin,Analytic functions of class H p, Trans. Am. Math. Soc.78 (1955), 46–66. · Zbl 0064.31203  M. Schiffer,Half-order differentials on Riemann surfaces, SIAM J. Appl. Math.14 (1966), 922–934. · Zbl 0164.37903  M. Schiffer and N. S. Hawley,Half-order differentials on Riemann surfaces, Acta Math.115 (1966), 199–236. · Zbl 0136.06701  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.  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.  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).  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  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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.