Cyclic permutations of sequences and uniform partitions.

Huang, Po-Yi; Ma, Jun; Yeh, Yeong-Nan

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Let $α > 0$ , $λ = {(2 α)}^{- 1 / 2}$ , $S^{n - 1}$ be the $(n - 1)$ -dimensional unit sphere, $σ$ be the surface measure on $S^{n - 1}$ and $h (x) = \int_{S^{n - 1}} e^{λ 〈 x, y 〉} d σ (y)$ . We characterize all subsets $M$ of $ℝ^{n}$ such that $inf_{x \in ℝ^{n}} \frac{u (x)}{h (x)} = inf_{x \in M} \frac{u (x)}{h (x)}$ for every positive solution $u$ of the Helmholtz equation on $ℝ^{n}$ . A closely related problem of representing functions of $L_{1} (S^{n - 1})$ as sums of blocks of the form $e^{λ 〈 x_{k}, . 〉} / h (x_{k})$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.