Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus
Steven B. Damelin, Joel Nathe
TL;DR
The paper extends energy minimization from classical logarithmic/Riesz settings to general subharmonic kernels on compact metric spaces. It proves that for regular entirely subharmonic kernels, minimizers are exactly those with constant potential outside a capacity-zero set (approximately $K$-invariant), using a first maximum principle. It then applies the theory to group-invariant kernels on compact homogeneous manifolds, showing that the Haar (uniform) measure minimizes energy in this setting, with concrete results for the flat torus: the Riesz kernel $K_s$ is minimized by the uniform measure when $d>s\ge d-2$, and minimizers have full support when $d>s>d-2$. The work also ties positive definiteness to minimization via a Ninomiya-type argument and links harmonic-polynomial expansions to nonnegative coefficients, yielding nonnegative multivariate Fourier cosine coefficients for subharmonic kernels on the torus, with implications for spectral representations of associated functions.
Abstract
We study the minimization of the energy integral $I_K(μ) = \int_Ω \int_Ω K(x,y) dμ(x) dμ(y)$ over all Borel probability measures $μ$, where $(Ω,ρ)$ is a compact connected metric space and $K:Ω^2 \to [0,\infty]$ is continuous in the extended sense. We focus on kernels $K$ which are subharmonic, which we define so that the potential $U_K^μ(x) = \int_Ω K(x,y) dμ(y)$ satisfies a maximum principle on $Ω\setminus{\rm supp}μ$. This extends the classical electrostatics minimization problem for logarithmic energy $\int_Ω\int_Ω\log\left(\frac{1}{||x-y||}\right)$, which is used heavily as a tool in approximation theory. Using properties of minimizing measures, we show that if the singularities of the subharmonic kernel $K$ are such that $K$ is regular, then $K$ is positive definite, and $μ$ is a minimizing measure if and only if its potential is constant (outside of a small exceptional set).We then apply this result to group invariant kernels on compact homogeneous manifolds. In this case, the uniform measure $σ$ has constant potential, so subharmonicity implies that this is the minimizing measure. Finally, we look at the case of the $d$-dimensional flat torus $T^d$. We use our results to see that the Riesz kernel $K_s(x,y) = {\rm sign}(s)ρ(x,y)^{-s}$ is minimized by $σ$ (and thus positive definite) when $d > s \geq d-2$. Additionally, the positive definiteness gives us a condition which implies that the multivariate Fourier series of a function $f:[0,π]^d \to [0,\infty]$ has nonnegative coefficients.