Covering and separation of Chebyshev points for non-integrable Riesz\n potentials

2017/02/28 by Alexander Reznikov, Edward B. Saff, Reznikov, Alexander +3
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1703.00106

openalex publication_date 2017/02/28 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

For Riesz s-potentials K(x,y)=|x-y|^-s, s>0, we investigate\nseparation and covering properties of N-point configurations\n\ω^*_N= x₁, \…, x_N on a d-dimensional compact set A\⊂\n\ℝ^\ℓ for which the minimum of \∑_j=1^N K(x, x_j) is maximal.\nSuch configurations are called N-point optimal Riesz s-polarization (or\nChebyshev) configurations. For a large class of d-dimensional sets A we\nshow that for s>d the configurations \ω^*_N have the optimal order of\ncovering. Furthermore, for these sets we investigate the asymptotics as N\→\n\∞ of the best covering constant. For these purposes we compare\nbest-covering configurations with optimal Riesz s-polarization configurations\nand determine the s-th root asymptotic behavior (as s\→ \∞) of the\nmaximal s-polarization constants. In addition, we introduce the notion of\n"weak separation" for point configurations and prove this property for optimal\nRiesz s-polarization configurations on A for s>dim(A), and for\nd-1 leqslant s < d on the sphere mathbbS^d.\n

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