Quadrature and harmonic 𝐿¹-approximation in annuli

Armitage, D. H.; Goldstein, M.

doi:10.1090/S0002-9947-1989-0949896-5

Quadrature and harmonic $L^ 1$-approximation in annuli
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by D. H. Armitage and M. Goldstein

Trans. Amer. Math. Soc. 312 (1989), 141-154

DOI: https://doi.org/10.1090/S0002-9947-1989-0949896-5

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Abstract:

Open sets $D$ in ${R^N}\;(N \geq 3)$ with the property that $\bar D$ is a closed annulus $\{ x:{r_1} \leq \;\left \| x\right \| \; \leq {r_2}\}$ are characterized by quadrature formulae involving mean values of certain harmonic functions. One such characterization is used to give a criterion for the existence of a best harmonic ${L^1}$ approximant to a function which is subharmonic (and satisfies some other conditions) in an annulus.

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