Abstract
Let h be a harmonic function on ℝn of suitably restricted growth. It is known that if h vanishes, or is bounded, on the lattice ℤn−1 × {0}, then the same is true on ℝn−1 × {0}. This paper presents sharp results which show that, if n [ges ] 3, then the same conclusions can be drawn even if information about h is missing on a substantial proportion of the lattice points. As corollaries we obtain uniqueness and Liouville-type theorems for harmonic, and also polyharmonic, functions which improve results by several authors.
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