×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. S. Azarin,On rays of completely regular growth of an entire function, Math. USSR Sb.8 (1969), 437–450. · Zbl 0197.35502 · doi:10.1070/SM1969v008n04ABEH001123
[2] A. Beurling inThe Collected Works of Arne Beurling (L. Carleson, P. Malliavin, J. Neuberger and J. Wermer, eds.), Birkhäuser, Boston, 1989, pp. 341–365.
[3] A. Beurling and P. Malliavin,On Fourier transforms of measures with compact support, Acta Math.107 (1962), 291–309. · Zbl 0127.32601 · doi:10.1007/BF02545792
[4] M. Christ, On the \(\bar \partial \) equation in weighted L2 norms in \(\bar \partial \) , J. Geom. Anal.1 (1991), 193–230. · Zbl 0737.35011
[5] D. Drasin,Approximation of subharmonic functions with applications, preprint, Montréal, 2000.
[6] A. Goldberg and M. Himyk,Approximation of subharmonic functions by logarithms of moduli of entire functions in integral metrics, preprint, 1999.
[7] A. F. Grishin, Private communication, 1975.
[8] W. K. Hayman,Subharmonic Functions, Vol. 2, Academic Press, London, New York, 1989. · Zbl 0699.31001
[9] W. Al-Katifi,On the asymptotic values and paths of certain integral and meromorphic functions, Proc. London. Math. Soc.16 (1966), 599–634. · Zbl 0145.30801 · doi:10.1112/plms/s3-16.1.599
[10] P. B. Kennedy,A class of integral functions bounded on certain curves, Proc. London. Math. Soc.6 (1956), 518–547. · Zbl 0074.29903 · doi:10.1112/plms/s3-6.4.518
[11] B. Kjellberg,On Certain Integral and Harmonic Functions. A Study in Minimum Modulus, Thesis, University of Uppsala, 1948. · Zbl 0031.16003
[12] J. Korevaar and M. A. Monterie,Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy, Trans. Amer. Math. Soc.350 (1998), 2329–2348. · Zbl 0892.31006 · doi:10.1090/S0002-9947-98-02187-4
[13] B. Ja. Levin,Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1980.
[14] Yu. Lyubarskii and K. Seip,Sampling and interpolation of entire functions and exponential systems in convex domains, Ark. Mat.32 (1994), 157–193. · Zbl 0819.30021 · doi:10.1007/BF02559527
[15] Yu. I. Lyubarskii and M. L. Sodin,Analogues of Sine Type Function for Convex Domains, Preprint no. 17, Institute for Low Temperatures, Ukrainian Acad. Sci., Kharkov, 1986 (Russian).
[16] S. N. Mergeljan,On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. Ser. 219 (1962), 109–166. · Zbl 0122.31601
[17] F. Nazarov, S. Treil and A. Volberg,Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices no. 15 (1997), 703–726. · Zbl 0889.42013 · doi:10.1155/S1073792897000469
[18] J. Ortega-Cerdà, Multipliers and weighted \(\bar \partial \) estimates, preprint, 1999.
[19] J. Ortega-Cerdà and K. Seip,Beurling-type density theorems for weighted L p spaces of entire functions, J. Analyse Math.75 (1998), 247–266. · Zbl 0920.30039 · doi:10.1007/BF02788702
[20] Ortega-Cerdà and K. Seip,Multipliers for entire functions and an interpolation problem of Beurling, J. Funct. Anal.162 (1999), 400–415. · Zbl 0916.30036 · doi:10.1006/jfan.1998.3357
[21] E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[22] R. S. Yulmukhametov,Approximation of subharmonic functions, Anal. Math.11 (1985), no. 3, 257–282 (Russian). · Zbl 0594.31005 · doi:10.1007/BF01907421
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.