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Riesz Energy on the Torus: Regularity of Minimizers Lu, Jianfeng Steinerberger, Stefan 2017-11-30T21:57:22Z 2017-11-30T21:57:22Z 2017-11-30
dc.description.abstract We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for $0< s < d$, the minimizing configuration of points has optimal regularity w.r.t. a Fourier-analytic regularity measure that arises in the study of irregularities of distribution. A particular consequence is that they are optimal quadrature points in the space of trigonometric polynomials up to a certain degree. The proof extends to other settings and also covers less singular functions such as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.
dc.subject math-ph
dc.subject math-ph
dc.subject math.CA
dc.subject math.MP
dc.title Riesz Energy on the Torus: Regularity of Minimizers
dc.type Journal article Lu, Jianfeng|0598771
pubs.organisational-group Chemistry
pubs.organisational-group Duke
pubs.organisational-group Mathematics
pubs.organisational-group Physics
pubs.organisational-group Temp group - logins allowed
pubs.organisational-group Trinity College of Arts & Sciences
duke.contributor.orcid Lu, Jianfeng|0000-0001-6255-5165

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