Restricted mean values and harmonic functions

Baxter, John R.

doi:10.1090/S0002-9947-1972-0293112-4

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Restricted mean values and harmonic functions
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by John R. Baxter

Trans. Amer. Math. Soc. 167 (1972), 451-463

DOI: https://doi.org/10.1090/S0002-9947-1972-0293112-4

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Abstract:

A function h defined on a region R in ${{\mathbf {R}}^n}$ will be said to possess a restricted mean value property if the value of the function at each point is equal to the mean value of the function over one open ball in R, with centre at that point. It is proved here that this restricted mean value property implies h is harmonic under certain conditions.

References

M. A. Akcoglu and R. W. Sharpe, Ergodic theory and boundaries, Trans. Amer. Math. Soc. 132 (1968), 447–460. MR 224770, DOI 10.1090/S0002-9947-1968-0224770-7