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Functions possessing restricted mean value properties


Author: David Heath
Journal: Proc. Amer. Math. Soc. 41 (1973), 588-595
MSC: Primary 31B05
DOI: https://doi.org/10.1090/S0002-9939-1973-0333213-1
MathSciNet review: 0333213
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Abstract: A real-valued function $ f$ defined on an open subset of $ {R^N}$ is said to have the restricted mean value property with respect to balls (spheres) if, for each point $ x$ in the set, there exists a ball (sphere) with center $ x$ and radius $ r(x)$ such that the average value of $ f$ over the ball (sphere) is equal to $ f(x)$. If $ f$ is harmonic then it has the restricted mean value property. Here new conditions for the converse implication are given.


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  • [1] M. A. Ackoglu and R. W. Sharpe, Ergodic theory and boundaries, Trans. Amer. Math. Soc. 132 (1968), 447-460. MR 37 #369. MR 0224770 (37:369)
  • [2] J. R. Baxter, Restricted mean values and harmonic functions, Trans. Amer. Math. Soc. 167 (1972), 451-463. MR 45 #2191. MR 0293112 (45:2191)
  • [3] K. L. Chung, The general theory of Markov processes according to Doblin, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 203-223. MR 0166835 (29:4108)
  • [4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience, New York, 1962. MR 25 #4216. MR 0140802 (25:4216)
  • [5] E. B. Dynkin, Markov processes. Vol. I, Fizmatgiz, Moscow, 1963; English transl., Die Grundlehren der math. Wissenschaften, Band 121, Academic Press, New York; Springer-Verlag, Berlin, 1965. MR 33 #1886. MR 0193670 (33:1886)
  • [6] H. Föllmer, Ein Littlewood-Kriterium für gleichmässig integrable Martingale und insbesondere für Dirichlet-Lösungen, Elliptische Differentialgleichungen, Band II, Akademie-Verlag, Berlin, 1971. MR 0326834 (48:5176)
  • [7] W. A. Veech, The core of a measurable set and a problem in potential theory (preprint).
  • [8] -, A converse to Gauss' theorem, Bull. Amer. Math. Soc. 78 (1972), 444-446. MR 0289800 (44:6987)
  • [9] -, A zero-one law for a class of random walks and a converse to Gauss' mean value theorem (preprint).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0333213-1
Keywords: Harmonic function, mean value property, Markov process
Article copyright: © Copyright 1973 American Mathematical Society

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