Riesz kernel energy

Builds on Borel measure Euclidean distance Riesz potential

Definition

Let $d \ge 1$ be an integer and let $s > 0$ . The Riesz $s$ -kernel on $\R^d$ is $k_s(x, y) = \abs{x - y}^{-s}$ for $x \ne y$ , and $k_s(x, x) = +\infty$ , where $\abs{x - y}$ is the Euclidean distance. Let $\mu$ be a finite positive Borel measure on $\R^d$ with compact support. The Riesz $s$ -energy of $\mu$ is $I_s(\mu) = \int \int \frac{d\mu(x)\, d\mu(y)}{\abs{x - y}^{s}} \in [0, +\infty].$ The measure $\mu$ has finite $s$ -energy if $I_s(\mu) < \infty$ . For distinct points $x_1, \dots, x_N \in \R^d$ , the discrete Riesz $s$ -energy is $E_s(x_1, \dots, x_N) = \sum_{i \ne j} \frac{1}{\abs{x_i - x_j}^{s}}.$

Remarks

The inner integral $\int \abs{x - y}^{-s}\, d\mu(y)$ is, up to a constant factor when $0 < s < d$ , the Riesz potential of $\mu$ at $x$ . So $I_s(\mu)$ is the integral of that potential against $\mu$ . Example: let $d = 3$ , $s = 1$ and let $\sigma$ be the uniform probability measure on the unit sphere. By the shell theorem, $\int \abs{x - y}^{-1}\, d\sigma(y) = 1$ for every $x$ on the sphere, so $I_1(\sigma) = 1$ . Non-example of finite energy: a point mass $\delta_a$ has $I_s(\delta_a) = k_s(a, a) = +\infty$ for every $s > 0$ . For $s \ge d$ , every nonzero measure as above has infinite $s$ -energy. The Riesz $s$ -capacity of a compact set $K$ is $1 / \inf I_s(\mu)$ , the infimum over Borel probability measures $\mu$ on $K$ .

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Tag	`7PN328ANQA`
URI	`m2://7PN328ANQA`
Kind	definition
Writing	W0 stub
MSC	31B15, 31C15
Defines	Riesz kernel energy
Uses	Borel measure, Euclidean distance, Riesz potential

Riesz kernel energy

Definition

Remarks

Reading path

Further reading

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