Approximation of Subharmonic Functions in the Unit Disk - Chyzhykov, I. E. (Журнал математичної фізики, аналізу, геометрії, 2008)
We prove that if u is a subharmonic function in D={|z|<1}, then there exists an absolute constant C and an analytic function f in D such that _D |u(z)-|f(z)||dm(z)<C, where m denotes the plane Lebesgue measure. We also (following the arguments of Lyubarskii and Malinnikova) answer Sodin’s question, namely, we show that the logarithmic potential of measure μ supported in a square Q, with μ(Q) being an integer N, admits approximations by the subharmonic function |P(z)|, where P is a polynomial with _Q |U_μ(z)-|P(z)||dxdy=O(1), independent of N and μ. We also consider uniform approximations. Mathematics Subject Classification: 31A05, 30E10.