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Characterization of best harmonic and superharmonic $$L^ 1$$-approximants. (English) Zbl 0853.31002
The authors give a complete solution to the problem of characterizing a best harmonic $$L^1$$-approximant $$h^*$$ to a function $$s$$ which is subharmonic on the open unit ball $$B$$ in $$\mathbb{R}^n$$, $$n\geq 2$$, and continuous on the closed unit ball $$\overline {B}$$. Such an approximant $$h^*$$, when it exists, is unique and equal to the solution of a certain Dirichlet problem. The technique is different from that used in earlier papers on this topic which imposed unduly restrictive hypotheses requiring $$s$$ to be smooth and not harmonic on any open set. The corresponding problem for superharmonic $$L^1$$-approximation of $$s$$ is also solved.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 41A30 Approximation by other special function classes 41A50 Best approximation, Chebyshev systems
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