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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the mean value property of harmonic functions and best harmonic $ L\sp 1$-approximation


Authors: Myron Goldstein, Werner Haussmann and Lothar Rogge
Journal: Trans. Amer. Math. Soc. 305 (1988), 505-515
MSC: Primary 31B05; Secondary 41A30, 41A50
MathSciNet review: 924767
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Abstract: The present paper deals with the inverse mean value property of harmonic functions and with the existence, uniqueness, and characterization of a best harmonic $ {L^1}$-approximant to strictly subharmonic functions. The main theorem concerning the inverse mean value property of harmonic functions is based on a generalization of a theorem due to Ü. Kuran as well as on an approximation theorem proved by J. C. Polking and also by L. I. Hedberg. The inverse mean value property will be applied in order to prove necessary and sufficient conditions for the existence of a best harmonic $ {L^1}$-approximant to a subharmonic function $ s$ satisfying $ \Delta s > 0$ a.e. in the unit ball.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0924767-8
PII: S 0002-9947(1988)0924767-8
Keywords: Harmonic and subharmonic functions, inverse mean value property, harmonic $ {L^1}$-approximation
Article copyright: © Copyright 1988 American Mathematical Society