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Covering and separation of Chebyshev points for non-integrable Riesz potentials | | Alexander Reznikov
; Edward B. Saff
; Alexander Volberg
; | | Date: |
1 Mar 2017 | | Abstract: | For Riesz $s$-potentials $K(x,y)=|x-y|^{-s}$, $s>0$, we investigate
separation and covering properties of $N$-point configurations
$omega^*_N={x_1, ldots, x_N}$ on a $d$-dimensional compact set $Asubset
mathbb{R}^ell$ for which the minimum of $sum_{j=1}^N K(x, x_j)$ is maximal.
Such configurations are called $N$-point optimal Riesz $s$-polarization (or
Chebyshev) configurations. For a large class of $d$-dimensional sets $A$ we
show that for $s>d$ the configurations $omega^*_N$ have the optimal order of
covering. Furthermore, for these sets we investigate the asymptotics as $N o
infty$ of the best covering constant. For these purposes we compare
best-covering configurations with optimal Riesz $s$-polarization configurations
and determine the $s$-th root asymptotic behavior (as $s o infty$) of the
maximal $s$-polarization constants. In addition, we introduce the notion of
"weak separation" for point configurations and prove this property for optimal
Riesz $s$-polarization configurations on $A$ for $s> ext{dim}(A)$, and for
$d-1leqslant s < d$ on the sphere $mathbb{S}^d$. | | Source: | arXiv, 1703.0106 | | Services: | Forum | Review | PDF | Favorites | |
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