Approximation of subharmonic functions with applications.

*(English)*Zbl 0991.31001
Arakelian, N. (ed.) et al., Approximation, complex analysis, and potential theory. Proceedings of the NATO Advanced Study Institute on modern methods in scientific computing and applications, MontrĂ©al, Canada, July 3-14, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 37, 163-189 (2001).

If \(f\) is an entire function on \(\mathbb{C}\) and \(f\not\equiv 0\), then \(\log|f|\) is subharmonic. The article under review discusses the approximation of general subharmonic functions by those of the special form \(\log|f|\).

Such approximations are achieved by first approximating the Riesz measure \(\mu\) of a subharmonic function \(u\) by a measure \(\nu\) consisting of unit point masses at a discrete set of points. (Then \(\nu\) becomes the Riesz measure of the approximating function \(\log|f|\).) An important preliminary result concerns the decomposition of a compactly supported measure with integer total mass as a sum of unit measures whose supports have a number of special properties. The decomposition theorem is based on the work of R. Yulmukhametov [Anal. Math. 11, 257-282 (1985; Zbl 0594.31005)] but the formulation is new and so are parts of the proof, which is given in detail.

If \(u\) is a subharmonic function of finite order on \(\mathbb{C}\), then there is an entire function \(r\) such that \(|u(z)-\log|f(x)||= O(\log|z|)\) as \(z\to\infty\) with \(z\) outside a certain exceptional set. A theorem of this kind, again based on Yulmukhametov’s work, is stated and proved. Some recent results on mean approximation, valid for functions of unrestricted growth, are also presented. One of these, due to Yu. Lyubarskii and E. Malinnikova [J. Math. Anal. 83, 121-149 (2001; Zbl 0981.31002)] asserts that if \(u\) is subharmonic on \(\mathbb{C}\) and \(q> 1/2\), then there exist \(R_0> 0\) and an entire function \(f\) such that \[ (\pi R^2)^{-1} \int_{|z|<R}|u(z)- \log|f(z)||dm_z< q\log R \] when \(R>R_0\). This fails if \(q< 1/2\).

Applications of approximation theorems to the theory of Levin-Pfluger functions (entire functions of completely regular growth) and to meromorphic functions with prescribed asymptotic values are discussed.

For the entire collection see [Zbl 0972.00045].

Such approximations are achieved by first approximating the Riesz measure \(\mu\) of a subharmonic function \(u\) by a measure \(\nu\) consisting of unit point masses at a discrete set of points. (Then \(\nu\) becomes the Riesz measure of the approximating function \(\log|f|\).) An important preliminary result concerns the decomposition of a compactly supported measure with integer total mass as a sum of unit measures whose supports have a number of special properties. The decomposition theorem is based on the work of R. Yulmukhametov [Anal. Math. 11, 257-282 (1985; Zbl 0594.31005)] but the formulation is new and so are parts of the proof, which is given in detail.

If \(u\) is a subharmonic function of finite order on \(\mathbb{C}\), then there is an entire function \(r\) such that \(|u(z)-\log|f(x)||= O(\log|z|)\) as \(z\to\infty\) with \(z\) outside a certain exceptional set. A theorem of this kind, again based on Yulmukhametov’s work, is stated and proved. Some recent results on mean approximation, valid for functions of unrestricted growth, are also presented. One of these, due to Yu. Lyubarskii and E. Malinnikova [J. Math. Anal. 83, 121-149 (2001; Zbl 0981.31002)] asserts that if \(u\) is subharmonic on \(\mathbb{C}\) and \(q> 1/2\), then there exist \(R_0> 0\) and an entire function \(f\) such that \[ (\pi R^2)^{-1} \int_{|z|<R}|u(z)- \log|f(z)||dm_z< q\log R \] when \(R>R_0\). This fails if \(q< 1/2\).

Applications of approximation theorems to the theory of Levin-Pfluger functions (entire functions of completely regular growth) and to meromorphic functions with prescribed asymptotic values are discussed.

For the entire collection see [Zbl 0972.00045].

Reviewer: D.H.Armitage (Belfast)

##### MSC:

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |

30D15 | Special classes of entire functions of one complex variable and growth estimates |