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Successive averages and harmonic functions. (English) Zbl 0879.31002
For every $$r>0$$ let $$m_r$$ be the normalized Lebesgue measure on the ball $$B(0, r)$$ in $$\mathbb{R}^d$$. This paper is motivated by the following question asked by G. Choquet in 1994: Let $$f$$ be a continuous real function on $$\mathbb{R}^d$$ and let $$r_1$$, $$r_2$$, $$r_3$$, $$\dots$$ be strictly positive real numbers; under what conditions on $$f$$ and the sequence $$(r_n)$$ does $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converge to a harmonic function?
Among other things, the authors prove the following result. If $$\sum^{\infty}_{n=1}r^2_n=\infty$$, then the following holds: (1) for every superharmonic function $$u$$ on $$\mathbb{R}^d$$ admitting a harmonic minorant the sequence $$(u\ast m_{r_1}\ast m_{r_2} \ast\cdots\ast m_{r_n})$$ decreases locally uniformly to the greatest harmonic minorant of $$u$$; (2) for every continuous $$f$$ with comapct support the sequence $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converges uniformly to zero; and (3) for every bounded $$f$$ such that the average $$\int fdm_r$$ tends to some real number $$A$$ as $$r\rightarrow\infty$$, the sequence $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converges locally uniformly to $$A$$.
The paper also contains similar results when $$m_r$$ is replaced by more general measures.
Reviewer: R.Song (Ann Arbor)

MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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References:
  H. Bauer,Probability Theory, de Gruyter, Berlin, 1996.  Y. Benyamini and Y. Weit,Functions satisfying the mean value property in the limit, J. Analyse Math.52 (1989), 167–198. · Zbl 0675.31002  J. Bliedtner and W. Hansen,Potential Theory–An Analytic and Probabilistic Approach to Balayage, Springer, Berlin, Heidelberg, New York, Tokyo, 1986. · Zbl 0706.31001  G. Choquet,Successive averages, inPotential Theory–ICPT 94, Proceedings of the International Conference on Potential Theory held in Kouty, Czech Republic, August 13–20, 1994, Walter de Gruyter, Berlin, 1996, p. 478.
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