Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915819089297408 |
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| author | Damelin, Steven B. Nathe, Joel |
| author_facet | Damelin, Steven B. Nathe, Joel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus Damelin, Steven B. Nathe, Joel Classical Analysis and ODEs 31B05, 31C05, 31A15, 30C85, 47B34, 30C40, 43A85 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. |
| title | Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus |
| topic | Classical Analysis and ODEs 31B05, 31C05, 31A15, 30C85, 47B34, 30C40, 43A85 |
| url | https://arxiv.org/abs/2410.01489 |