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

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 $f(x) = \exp\bigl(- N^{\frac{2}{d}} |x|^2 \bigr)$.

http://arxiv.org/abs/1710.08010v1

https://hdl.handle.net/10161/15781

math-ph

math-ph

math.CA

math.MP

Riesz Energy on the Torus: Regularity of Minimizers

Journal article

Lu, Jianfeng|0000-0001-6255-5165

http://arxiv.org/abs/1710.08010v1

Chemistry

Duke

Mathematics

Physics

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Trinity College of Arts & Sciences