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The convergence of polynomial expansions of positive harmonic functions

Published online by Cambridge University Press:  24 October 2008

D. H. Armitage
D. H. Armitage
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN
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Extract

Let B(r) denote the open ball of radius r centred at the origin 0 of the Euclidean space ℝN, where N ≥ 2. It is well known that if h is harmonic in B(1), then there exist homogeneous harmonic polynomials Hj of degree j in ℝN such that converges absolutely and locally uniformly to h in B(1) (see, e.g. Brelot[1], Appendice). Further, this series is unique and each Hj is the sum of all the monomial terms of degree j in the multiple Taylor series of h centred at 0. We call the polynomial expansion of h.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 165 - 174
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Brelot, M.. Éléments de la théorie classique du potentiel (Centre de documentation universitaire, Paris, 1969).Google Scholar
[2]Doob, J. L.. Classical Potential Theory and Its Probabilistic Counterpart (Springer, 1983).Google Scholar
[3]Fejér, L.. Über die Laplacesche Reihe. Math. Ann. 67 (1909), 76–109.CrossRefGoogle Scholar
[4]Fejér, L., Über die Positivität von Summen, die nach trigonometrischen oder Legendreschen Funktionen fortschreiten (Erste Mitteilung), Acta litterarum ac scientiarum regiae universitatis hungaricae Francisco-Josephinae. secto scientiarum mathematicarum 2 (1925), 75–86.Google Scholar
[5]Feldheim, E.. On the positivity of certain sums of ultraspherical polynomials. J. d'Analyse Math. 11 (1963), 275–284.CrossRefGoogle Scholar
[6]Müller, C.. Spherical Harmonics (Lecture Notes in Math, vol. 17, Springer, 1966).CrossRefGoogle Scholar
[7]Szegö, G.. Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Ent-wicklungen. Math. Ann. 96 (1927), 601–632.CrossRefGoogle Scholar
[8]Szegö, G.. Orthogonal Polynomials (American Mathematical Society, 1967).Google Scholar