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Approximation by rational functions on Riemann surfaces


Authors: M. Goldstein and J. L. Walsh
Journal: Proc. Amer. Math. Soc. 36 (1972), 464-466
MSC: Primary 30A82
DOI: https://doi.org/10.1090/S0002-9939-1972-0313518-X
MathSciNet review: 0313518
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Abstract: In this paper, we show that if $ F \in {L^p}(k,\alpha )$ on $ \Gamma $ where $ \Gamma $ denotes the border of a compact bordered Riemann surface $ \bar R$, then F can be uniquely written as the sum of a function in $ {H^p}(k,\alpha )$ and a function in $ {G^p}(k,\alpha )$ and moreover that F can be approximated on $ \Gamma $ in $ {L^p}$ norm to within $ A/{n^{k + \alpha }}$ by a sequence of rational functions on the union of $ \bar R$ with its double.


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  • [2] M. Heins, Hardy classes on Riemann surfaces, Lecture Notes in Math., no. 98, Springer-Verlag, Berlin and New York, 1969. MR 40 #338. MR 0247069 (40:338)
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  • [4] J. L. Walsh, Mean approximation by polynomials on a Jordan curve, J. Approximation Theory 4 (1971), 263-268. MR 0294658 (45:3726)

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DOI: https://doi.org/10.1090/S0002-9939-1972-0313518-X
Keywords: Lipschitz condition, Hardy classes, rational functions, meromorphic differential
Article copyright: © Copyright 1972 American Mathematical Society

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