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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.

##### MSC:
 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
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