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Universal polynomial expansions of harmonic functions. (English) Zbl 1271.31003
Let \(h\) be a harmonic function on a domain \(\Omega \subset \mathbb{R}^{N}\) and let \(w\in \Omega \). We say that \(h\in U(\Omega ,w)\) if, on each compact \(K\subset \mathbb{R}^{N}\backslash \Omega \) with connected complement, arbitrary harmonic polynomials can be uniformly approximated by suitable partial sums of the homogeneous polynomial expansion of \(h\) about \(w\). It is known that \(U(\Omega ,w)\neq \emptyset \) if \((\mathbb{R}^{N}\cup \{\infty \})\backslash \Omega \) is connected. The author shows, among other things, that, provided \(\Omega \) omits an infinite cone, (1) \(U(\Omega ,w)\neq \emptyset \) if and only if \((\mathbb{R}^{N}\cup \{\infty \})\backslash \Omega \) is connected; (2) the collection \(U(\Omega ,w)\) is independent of the choice of \(w\); (3) no function in \(U(\Omega ,w)\) has a harmonic extension to a larger domain. The paper includes, and exploits, a careful analysis of the gap structure of the homogeneous polynomial expansion of functions belonging to \(U(\Omega ,w)\) for domains \(\Omega \) that omit a cone.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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[1] Armitage, DH, Universal overconvergence of polynomial expansions of harmonic functions, J. Approx. Theory, 118, 225-234, (2002) · Zbl 1374.31002
[2] Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London (2001) · Zbl 0972.31001
[3] Bacharoglou, A; Stamatiou, G, Universal harmonic functions on the hyperbolic space, Colloq. Math., 121, 93-105, (2010) · Zbl 1206.31004
[4] Bayart, F; Grosse-Erdmann, K-G; Nestoridis, V; Papadimitropoulos, C, Abstract theory of universal series and applications, Proc. Lond. Math. Soc. (3), 96, 417-463, (2008) · Zbl 1147.30003
[5] Gauthier, P; Tamptse, I, Universal overconvergence of homogeneous expansions of harmonic functions, Analysis, 26, 287-293, (2006) · Zbl 1132.31002
[6] Hayman, WK, Power series expansions for harmonic functions, Bull. Lond. Math. Soc., 2, 152-158, (1970) · Zbl 0201.43302
[7] Klimek, M.: Pluripotential Theory. Oxford Science, Oxford, UK (1991) · Zbl 0742.31001
[8] Korevaar, J; Meyers, JLH, Logarithmic convexity for supremum norms of harmonic functions, Bull. Lond. Math. Soc., 26, 353-362, (1994) · Zbl 0819.31001
[9] Luh, W, Universal approximation properties of overconvergent power series on open sets, Analysis, 6, 191-207, (1986) · Zbl 0589.30003
[10] Manolaki, M, Ostrowski-type theorems for harmonic functions, J. Math. Anal. Appl., 391, 480-488, (2012) · Zbl 1238.31005
[11] Melas, A; Nestoridis, V, Universality of Taylor series as a generic property of holomorphic functions, Adv. Math., 157, 138-176, (2001) · Zbl 0985.30023
[12] Müller, J; Yavrian, A, On polynomial sequences with restricted growth near infinity, Bull. Lond. Math. Soc., 34, 189-199, (2002) · Zbl 1020.30044
[13] Müller, J; Vlachou, V; Yavrian, A, Universal overconvergence and Ostrowski-gaps, Bull. Lond. Math. Soc., 38, 597-606, (2006) · Zbl 1099.30001
[14] Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge, UK (1995) · Zbl 0828.31001
[15] Tamptse, I.: Approximation of harmonic functions by universal overconvergent series. PhD thesis, Univ. de Montreal (2008) · Zbl 1374.31002
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