The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background and Measurable Functions and the Lebesgue Integral: Standing Notation, and fix a dimension $q$. Points of $\mathbb{R}^{q}$ are read as $q$-tuples by Euclidean Points as Tuples of Real Numbers, the $i$th component of $z$ being $z_{i}$. Borel is as fixed in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, and for $A\in\mathcal{B}(\mathbb{R}^{q})$ the indicator $\mathbf{1}_{A}$ is the Borel function of claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. Let $\lambda_{q}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{q})$; integrals with respect to it are those of Measurable Functions and the Lebesgue Integral: Standing Notation (v), a nonnegative Borel function being integrated as a $[0,\infty]$-valued map, and $\int_{\mathbb{R}^{q}}f\,d\lambda_{q}$ is also written $\int_{\mathbb{R}^{q}}f(z)\,dz$; integrable means integrable with respect to $\lambda_{q}$. Smoothness of a function $\mathbb{R}^{q}\to\mathbb{R}$ is that of Smooth Functions on an Open Subset of ℝⁿ on $\mathbb{R}^{q}$, which is open by claim 1 of Euclidean Space Is Open in Itself; $\delta_{ij}$ is the entry in row $i$ and column $j$ of the identity matrix $I_{q}$; $\sqrt{t}$ is the nonnegative square root of a nonnegative real number $t$; for a positive real number $s$ and a natural number $m$, $s^{-m/2}$ denotes the multiplicative inverse of the $m$th natural power $(\sqrt{s})^{m}$, and $s^{-1/2}z$ the scalar multiple of $z\in\mathbb{R}^{q}$ by $s^{-1/2}$; and the natural number $q$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background (i) where a real number is required.

For every real number $s$ with $0<s\le1$ let $g_{s}:\mathbb{R}^{q}\to\mathbb{R}$ be the Gaussian smoothing weight $\varphi_{\eta}$ of claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, read in dimension $m=q$ with $\eta=s$; that is,

$$g_{s}(z)=c_{s}\exp\Bigl(-\frac{\lVert z\rVert^{2}}{2s}\Bigr),$$

where $c_{s}$ is the unique positive real number with $\int_{\mathbb{R}^{q}}g_{s}\,d\lambda_{q}=1$. Then the following hold.

1. (Scaling)(i) For every $s$ with $0<s\le1$ and every $z\in\mathbb{R}^{q}$,

$$c_{s}=c_{1}\,s^{-q/2}\qquad\text{and}\qquad g_{s}(z)=s^{-q/2}\,g_{1}\bigl(s^{-1/2}z\bigr).$$

2. (Derivatives)(ii) For every $s$ with $0<s\le1$ the function $g_{s}$ is smooth on $\mathbb{R}^{q}$ and even, $g_{s}(-z)=g_{s}(z)$, and for all $z\in\mathbb{R}^{q}$ and $i,j,k\in[q]$,

$$\partial_{i}g_{s}(z)=-\frac{z_{i}}{s}\,g_{s}(z),\qquad \partial_{j}\partial_{i}g_{s}(z)=\Bigl(\frac{z_{i}z_{j}}{s^{2}}-\frac{\delta_{ij}}{s}\Bigr)g_{s}(z),$$ $$\partial_{k}\partial_{j}\partial_{i}g_{s}(z)=\Bigl(-\frac{z_{i}z_{j}z_{k}}{s^{3}}+\frac{\delta_{ij}z_{k}+\delta_{ik}z_{j}+\delta_{jk}z_{i}}{s^{2}}\Bigr)g_{s}(z).$$

3. (Uniform and square-integral bounds)(iii) There are nonnegative real numbers $A_{0},A_{1},A_{2},A_{3}$ and $B_{0},B_{1},B_{2},B_{3}$, depending only on $q$, such that for every $s$ with $0<s\le1$, every $z\in\mathbb{R}^{q}$ and all $i,j,k\in[q]$,

$$g_{s}(z)\le A_{0}\,s^{-q/2},\qquad |\partial_{i}g_{s}(z)|\le A_{1}\,s^{-(q+1)/2},\qquad |\partial_{j}\partial_{i}g_{s}(z)|\le A_{2}\,s^{-(q+2)/2},\qquad |\partial_{k}\partial_{j}\partial_{i}g_{s}(z)|\le A_{3}\,s^{-(q+3)/2},$$

and the squares of $g_{s}$, $\partial_{i}g_{s}$, $\partial_{j}\partial_{i}g_{s}$ and $\partial_{k}\partial_{j}\partial_{i}g_{s}$ are integrable, with

$$\int_{\mathbb{R}^{q}}g_{s}^{2}\,d\lambda_{q}\le B_{0}\,s^{-q/2},\qquad \int_{\mathbb{R}^{q}}(\partial_{i}g_{s})^{2}\,d\lambda_{q}\le B_{1}\,s^{-(q+2)/2},$$ $$\int_{\mathbb{R}^{q}}(\partial_{j}\partial_{i}g_{s})^{2}\,d\lambda_{q}\le B_{2}\,s^{-(q+4)/2},\qquad \int_{\mathbb{R}^{q}}(\partial_{k}\partial_{j}\partial_{i}g_{s})^{2}\,d\lambda_{q}\le B_{3}\,s^{-(q+6)/2}.$$

4. (Convolution identity)(iv) For every $s$ with $0<s\le\tfrac12$ and all $x,x'\in\mathbb{R}^{q}$, the function $y\mapsto g_{s}(y-x)\,g_{s}(y-x')$ on $\mathbb{R}^{q}$ is integrable and

$$\int_{\mathbb{R}^{q}}g_{s}(y-x)\,g_{s}(y-x')\,dy=g_{2s}(x-x').$$

5. (Moments and tails)(v) For every $s$ with $0<s\le1$ the functions $z\mapsto\lVert z\rVert^{2}g_{s}(z)$ and $z\mapsto\lVert z\rVert\,g_{s}(z)$ are integrable,

$$\int_{\mathbb{R}^{q}}\lVert z\rVert^{2}g_{s}(z)\,dz=q\,s,\qquad\int_{\mathbb{R}^{q}}\lVert z\rVert\,g_{s}(z)\,dz\le\sqrt{q\,s},$$

and for every positive real number $R$, the set $A_{R}=\{z\in\mathbb{R}^{q}:R<\lVert z\rVert\}$ being Borel as the preimage of the Borel set $(R,\infty)$ under the Borel map $z\mapsto\lVert z\rVert$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv),

$$\int_{\mathbb{R}^{q}}\mathbf{1}_{A_{R}}\,g_{s}\,d\lambda_{q}\le\frac{q\,s}{R^{2}} .$$