Approximation in the Mean by Analytic Functions

Let be a compact set in the plane, let have its usual meaning, and let be the subspace of functions analytic in the interior of . The problem studied in this paper is whether or not rational functions with poles off are dense in (or in in the case when has no interior). For <!-- MATH $1 \leqq p \leqq 2$ --> the problem has been settled by Bers and Havin. By a method which applies for <!-- MATH $1 \leqq p < \infty$ --> <img width="99" height="41" align="MIDDLE" border="0" src="images/img15.gif" alt="$ 1 \leqq p < \infty $"> we give new results for 2$"> which improve earlier results by Sinanjan. The results are given in terms of capacities.