2024/10/02 by Steven B. Damelin, Damelin, Steven B., Joel Nathe +1
Economics, Econometrics and Finance · Physics and Astronomy · #30C40 #30C85 #31A15 #31B05 #31C05 #43A85 #47B34 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2410.01489
openalex publication_date 2024/10/02 · openalex created_date 2024/10/29 · openalex updated_date 2026/07/28
We study the minimization of the energy integral IK(μ) = ∫Ω ∫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:Ω2 → [0,∞] is continuous in the extended sense. We focus on kernels K which are subharmonic, which we define so that the potential UKμ(x) = ∫Ω K(x,y) dμ(y) satisfies a maximum principle on Ω∖\rm suppμ. This extends the classical electrostatics minimization problem for logarithmic energy ∫Ω∫Ωlog((1)/(||x-y||)), 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 Td. We use our results to see that the Riesz kernel Ks(x,y) = \rm sign(s)ρ(x,y)-s is minimized by σ (and thus positive definite) when d > s ≥ d-2. Additionally, the positive definiteness gives us a condition which implies that the multivariate Fourier series of a function f:[0,π]d → [0,∞] has nonnegative coefficients.