Approximation in the mean by analytic functions

Hedberg, Lars Inge

doi:10.1090/S0002-9947-1972-0432886-6

Approximation in the mean by analytic functions
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by Lars Inge Hedberg

Trans. Amer. Math. Soc. 163 (1972), 157-171

DOI: https://doi.org/10.1090/S0002-9947-1972-0432886-6

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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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Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen

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