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.


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