Abstract
In this work, we develop a theory of approximating general vector fields on subsets of the sphere in ℝn by harmonic gradients from the Hardy space H p of the ball, 1<p<∞. This theory is constructive for p=2, enabling us to solve approximate recovery problems for harmonic functions from incomplete boundary values. An application is given to Dirichlet–Neumann inverse problems for n=3, which are of practical importance in medical engineering. The method is illustrated by two numerical examples.
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Communicated by Carlos Kenig.
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Atfeh, B., Baratchart, L., Leblond, J. et al. Bounded Extremal and Cauchy–Laplace Problems on the Sphere and Shell. J Fourier Anal Appl 16, 177–203 (2010). https://doi.org/10.1007/s00041-009-9110-0
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DOI: https://doi.org/10.1007/s00041-009-9110-0