The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, with Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation in force, fix a dimension $q$. Let $g_{s}$ for $0<s\le1$ be the Gaussian kernels of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails; let $g_{s}*\mu$ be the Gaussian smoothing of $\mu\in\mathcal{P}(\mathbb{R}^{q})$ at scale $s$, with Lebesgue measure $\lambda_{q}$, integrable and bounded as there; and let series of real numbers, their convergence, absolute convergence and sums be those of Series of Real Numbers. The third index of a third-order partial derivative is written $l$ here, the letter $k$ being the series index. For $k\in\mathbb{N}$ put

$$s_{k}=4^{-k}\qquad\text{and}\qquad w_{k}=2^{-k(q+8)},$$

the multiplicative inverses of the natural powers $4^{k}$ and $2^{k(q+8)}$, where $2=1+1$ and $4=2+2$; thus $0<s_{k}\le4^{-1}$ and $0<2s_{k}\le2^{-1}$ for every $k$. Then the following hold.

1. (The kernel)(i) For every $z\in\mathbb{R}^{q}$ the series $\sum_{k=1}^{\infty}w_{k}\,g_{2s_{k}}(z)$ converges absolutely. The function

$$K:\mathbb{R}^{q}\to\mathbb{R},\qquad K(z)=\sum_{k=1}^{\infty}w_{k}\,g_{2s_{k}}(z),$$

called the multiscale Gaussian kernel, is of class $C^{3}$ on $\mathbb{R}^{q}$ and even, $K(-z)=K(z)$; for all $z\in\mathbb{R}^{q}$ and $i,j,l\in[q]$ its partial derivatives are the absolutely convergent sums

$$\partial_{i}K(z)=\sum_{k=1}^{\infty}w_{k}\,\partial_{i}g_{2s_{k}}(z),\qquad \partial_{j}\partial_{i}K(z)=\sum_{k=1}^{\infty}w_{k}\,\partial_{j}\partial_{i}g_{2s_{k}}(z),\qquad \partial_{l}\partial_{j}\partial_{i}K(z)=\sum_{k=1}^{\infty}w_{k}\,\partial_{l}\partial_{j}\partial_{i}g_{2s_{k}}(z);$$

and there are nonnegative real numbers $M_{0},M_{1},M_{2},M_{3}$, depending only on $q$, such that for all $z\in\mathbb{R}^{q}$ and $i,j,l\in[q]$,

$$0\le K(z)\le M_{0},\qquad|\partial_{i}K(z)|\le M_{1},\qquad|\partial_{j}\partial_{i}K(z)|\le M_{2},\qquad|\partial_{l}\partial_{j}\partial_{i}K(z)|\le M_{3}.$$

2. (The potential of a measure)(ii) Let $\nu\in\mathcal{P}(\mathbb{R}^{q})$. For every $x\in\mathbb{R}^{q}$ the function $y\mapsto K(x-y)$ is bounded and Borel, so that

$$(K*\nu)(x)=\int_{\mathbb{R}^{q}}K(x-y)\,\nu(dy)$$

is a real number. The function $K*\nu:\mathbb{R}^{q}\to\mathbb{R}$, called the potential of $\nu$, is of class $C^{3}$ on $\mathbb{R}^{q}$, with

$$\partial_{i}(K*\nu)(x)=\int_{\mathbb{R}^{q}}\partial_{i}K(x-y)\,\nu(dy),\qquad \partial_{j}\partial_{i}(K*\nu)(x)=\int_{\mathbb{R}^{q}}\partial_{j}\partial_{i}K(x-y)\,\nu(dy),\qquad \partial_{l}\partial_{j}\partial_{i}(K*\nu)(x)=\int_{\mathbb{R}^{q}}\partial_{l}\partial_{j}\partial_{i}K(x-y)\,\nu(dy)$$

for all $x$ and $i,j,l\in[q]$, the integrands being bounded Borel functions of $y$; it satisfies $0\le K*\nu\le M_{0}$, $|\partial_{i}(K*\nu)|\le M_{1}$, $|\partial_{j}\partial_{i}(K*\nu)|\le M_{2}$ and $|\partial_{l}\partial_{j}\partial_{i}(K*\nu)|\le M_{3}$ on $\mathbb{R}^{q}$; and for every $x\in\mathbb{R}^{q}$ the series below converges absolutely and

$$(K*\nu)(x)=\sum_{k=1}^{\infty}w_{k}\,(g_{2s_{k}}*\nu)(x).$$

3. (The kernel pairing)(iii) Let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{q})$. The function $z\mapsto K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ on $\mathbb{R}^{q+q}$ is bounded and Borel, with the coordinate projections and product measure of that clause, and the kernel pairing

$$\mathcal{K}(\mu,\nu)=\int_{\mathbb{R}^{q+q}}K\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\bigr)\,(\mu\boxtimes\nu)(dz)$$

satisfies $0\le\mathcal{K}(\mu,\nu)\le M_{0}$, $\mathcal{K}(\mu,\nu)=\mathcal{K}(\nu,\mu)$, and

$$\mathcal{K}(\mu,\nu)=\int_{\mathbb{R}^{q}}K*\nu\,d\mu=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}(g_{s_{k}}*\mu)(g_{s_{k}}*\nu)\,d\lambda_{q},$$

where each product $(g_{s_{k}}*\mu)(g_{s_{k}}*\nu)$ is integrable and the series has nonnegative terms and converges.