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On approximation of subharmonic functions. (English) Zbl 0981.31002
Several authors have studied the approximation of subharmonic functions \(u\) in the plane by functions of the form \(\log|f|\) where \(f\) is an entire function. For example, V. S. Azarin [Math. USSR, Sb. 8, 437-450 (1969); translation from Mat. Sb., Nov. Ser. 79(121), 463-476 (1969; Zbl 0194.10701)] showed that if \(u\) has finite order \(\rho > 0\), then there is an entire function \(f\) such that \(u(z) - \log|f(z)|= o(|z|^\rho)\) as \(z\to\infty\) outside some exceptional set \(E\). R. S. Yulmukhametov [Anal. Math. 11, 257-282 (1985; Zbl 0594.31005)] subsequently showed that the error term “\(o(|z|^\rho)\)” can be replaced by “\(O(\log |z|)\)” for an appropriate choice of \(f\) and \(E\).
The present authors develop this line of work further by removing the assumption that \(u\) has finite order. They also carefully examine the sharpness of the error term and the structure and size of the exceptional set \(E\). Finally, an analogous result for approximation of Newtonian potentials in \(\mathbb{R}^n\) is obtained.

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30E10 Approximation in the complex plane
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
Full Text: DOI
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