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Transactions of the American Mathematical Society

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Approximation in the mean by analytic functions


Author: Lars Inge Hedberg
Journal: Trans. Amer. Math. Soc. 163 (1972), 157-171
MSC: Primary 30A82
DOI: https://doi.org/10.1090/S0002-9947-1972-0432886-6
MathSciNet review: 0432886
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Abstract: Let $ E$ be a compact set in the plane, let $ {L^p}(E)$ have its usual meaning, and let $ L_a^p(E)$ be the subspace of functions analytic in the interior of $ E$. The problem studied in this paper is whether or not rational functions with poles off $ E$ are dense in $ L_a^p(E)$ (or in $ {L^p}(E)$ in the case when $ E$ has no interior). For $ 1 \leqq p \leqq 2$ the problem has been settled by Bers and Havin. By a method which applies for $ 1 \leqq p < \infty $ we give new results for $ p > 2$ which improve earlier results by Sinanjan. The results are given in terms of capacities.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0432886-6
Keywords: Approximation in areal mean, rational functions, $ {L^p}$-spaces, compact sets, analytic $ p$-capacity
Article copyright: © Copyright 1972 American Mathematical Society

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