The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder

Let $m\ge1$ be a natural number, let $\mathbb{R}^m$ be Euclidean space with the Euclidean norm $\lVert\cdot\rVert$, the dot product $z\cdot w$, and the Euclidean distance $d$, and let $\lambda_m$ be Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^m)$. Let $\exp$ be the exponential function. A function $f:\mathbb{R}^m\to\mathbb{R}$ is called measurable if it is measurable with respect to $\mathcal{B}(\mathbb{R}^m)$ and the Borel $\sigma$-algebra of the real line, and sequentially continuous if $f(z^k)\to f(z)$ whenever $(z^k)_{k\in\mathbb{N}}$ is a sequence in $\mathbb{R}^m$ and $z\in\mathbb{R}^m$ with $d(z^k,z)\to0$, convergence being that of Limit of a Sequence of Real Numbers, as in claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Integrals of $[0,\infty]$-valued measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, integrable means integrable with respect to $\lambda_m$, and partial derivatives are taken on the open set $U=\mathbb{R}^m$ (open, since it contains every ball about any of its points). Sums and scalar multiples of points of $\mathbb{R}^m$ are those of Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$, so that $\mathbb{R}^m$ is a real vector space with negative $-z=0-z$. For a real $t\ge0$, $t^{1/2}=\sqrt{t}$ denotes the nonnegative square root, and $t^{-1/2}=1/t^{1/2}$ for $t>0$.

Let $\eta$ be a real number with $0<\eta\le1$ and define $\psi_\eta:\mathbb{R}^m\to\mathbb{R}$ by

$$\psi_\eta(z)=\exp\Bigl(-\frac{\lVert z\rVert^{2}}{2\eta}\Bigr).$$

1. (Normalization) $\psi_\eta$ is sequentially continuous and measurable, $0<\psi_\eta(z)\le1$ for every $z$, and $0<\int_{\mathbb{R}^m}\psi_\eta\,d\lambda_m<\infty$. Consequently there is exactly one real number $c_\eta>0$ with $\int_{\mathbb{R}^m}c_\eta\psi_\eta\,d\lambda_m=1$. The Gaussian smoothing weight $\varphi_\eta=c_\eta\psi_\eta$ is sequentially continuous and measurable, satisfies $0<\varphi_\eta(z)\le c_\eta$ and $\varphi_\eta(-z)=\varphi_\eta(z)$ for every $z$, and for every $a\in\mathbb{R}^m$ the map $z\mapsto\varphi_\eta(z-a)$ is measurable with

$$\int_{\mathbb{R}^m}\varphi_\eta(z-a)\,d\lambda_m(z)=1.$$

2. (Partial derivatives) For every $i\in\{1,\dots,m\}$ and every $z=(z_1,\dots,z_m)\in\mathbb{R}^m$ the partial derivative of $\varphi_\eta$ with respect to the $i$th variable exists at $z$ and

$$\partial_i\varphi_\eta(z)=-\frac{z_i}{\eta}\,\varphi_\eta(z).$$

The function $\partial_i\varphi_\eta$ is sequentially continuous and $|\partial_i\varphi_\eta(z)|\le c_\eta\,(2\eta)^{-1/2}$ for every $z$.

3. (Exponential tilting) Let $a\in\mathbb{R}^m$ and put $Z_a(z)=a\cdot z/\eta$ for $z\in\mathbb{R}^m$ and $\kappa_a=\lVert a\rVert^{2}/\eta$. Then for every $z\in\mathbb{R}^m$

$$\varphi_\eta(z)\,\exp\bigl(Z_a(z)\bigr)=\exp(\kappa_a/2)\,\varphi_\eta(z-a),$$

the functions $\varphi_\eta\exp(Z_a)$ and $z\mapsto\varphi_\eta(z-a)^{2}/\varphi_\eta(z)$ are measurable, and

$$\int_{\mathbb{R}^m}\varphi_\eta\exp(Z_a)\,d\lambda_m=\exp(\kappa_a/2),\qquad\int_{\mathbb{R}^m}\frac{\varphi_\eta(z-a)^{2}}{\varphi_\eta(z)}\,d\lambda_m(z)=\exp(\kappa_a).$$

4. (Moments) With $a$, $Z_a$ and $\kappa_a$ as in claim 3, the functions $Z_a\varphi_\eta$, $Z_a^{2}\varphi_\eta$ and $Z_a\exp(Z_a)\varphi_\eta$ are integrable, and

$$\int_{\mathbb{R}^m}Z_a\varphi_\eta\,d\lambda_m=0,\qquad\int_{\mathbb{R}^m}Z_a^{2}\varphi_\eta\,d\lambda_m=\kappa_a,\qquad\int_{\mathbb{R}^m}Z_a\exp(Z_a)\varphi_\eta\,d\lambda_m=\kappa_a\exp(\kappa_a/2).$$

5. (First-order remainder) With $a=(a_1,\dots,a_m)$, $Z_a$ and $\kappa_a$ as in claim 3, define $R_a:\mathbb{R}^m\to\mathbb{R}$ by

$$R_a(z)=\varphi_\eta(z-a)-\varphi_\eta(z)+\sum_{i=1}^{m}a_i\,\partial_i\varphi_\eta(z).$$

Then $R_a$ is sequentially continuous, $R_a(z)=\varphi_\eta(z)\bigl(\exp(Z_a(z)-\kappa_a/2)-1-Z_a(z)\bigr)$ for every $z$, the function $R_a^{2}/\varphi_\eta$ is integrable, and

$$\int_{\mathbb{R}^m}\frac{R_a(z)^{2}}{\varphi_\eta(z)}\,d\lambda_m(z)=\exp(\kappa_a)-1-\kappa_a\le\frac{\kappa_a^{2}}{2}\exp(\kappa_a).$$