Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence

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 $q$, and let $g_{s}$ for $0<s\le1$, and the notation $s^{-1/2}$, be those of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails; each $g_{s}$ is smooth on $\mathbb{R}^{q}$, so that its partial derivatives $\partial_{i}g_{s}$ exist, by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives. Lebesgue measure $\lambda_{q}$ on $\mathcal{B}(\mathbb{R}^{q})$ is that of Euclidean Space and Lebesgue Measure: Standing Notation §measure, integrals with respect to it are also written $\int f(z)\,dz$, and integrable means integrable with respect to $\lambda_{q}$. The class $C^{1}$ on $\mathbb{R}^{q}$ and partial derivatives are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, $\mathbb{R}^{q}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $C^{1}_{b}(\mathbb{R}^{q})$ is the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded. Let $H:\mathbb{R}^{q}\to\mathbb{R}$ be Borel and let $M$ be a nonnegative real number with $|H(z)|\le M$ for every $z\in\mathbb{R}^{q}$.

1. (Smoothing)¶ Let $0<s\le1$ and $y\in\mathbb{R}^{q}$. The functions $x\mapsto H(y-x)\,g_{s}(x)$ and $z\mapsto g_{s}(y-z)\,H(z)$ on $\mathbb{R}^{q}$ are Borel and integrable, and their integrals are equal; their common value is written

$$H_{s}(y)=\int_{\mathbb{R}^{q}}H(y-x)\,g_{s}(x)\,dx=\int_{\mathbb{R}^{q}}g_{s}(y-z)\,H(z)\,dz ,$$

and $|H_{s}(y)|\le M$.

2. (Derivatives through the kernel)¶ Let $0<s\le1$. The function $H_{s}:\mathbb{R}^{q}\to\mathbb{R}$ is of class $C^{1}$ on $\mathbb{R}^{q}$, and for every $i\in[q]$ and every $y\in\mathbb{R}^{q}$ the function $z\mapsto\partial_{i}g_{s}(y-z)\,H(z)$ is Borel and integrable, with

$$\partial_{i}(H_{s})(y)=\int_{\mathbb{R}^{q}}\partial_{i}g_{s}(y-z)\,H(z)\,dz,\qquad|\partial_{i}(H_{s})(y)|\le M\,\sqrt{q}\;s^{-1/2}.$$

3. (Derivatives through the function)¶ Suppose that $H\in C^{1}_{b}(\mathbb{R}^{q})$, and let $0<s\le1$ and $i\in[q]$. Then $\partial_{i}H$ is Borel and bounded, so that its smoothing $(\partial_{i}H)_{s}$ is defined by claim 1 applied with $\partial_{i}H$ and a bound of it in place of $H$ and $M$, and $\partial_{i}(H_{s})=(\partial_{i}H)_{s}$.

4. (Pointwise convergence)¶ Let $y\in\mathbb{R}^{q}$ be a point at which $H$ is continuous. For every positive $\varepsilon\in\mathbb{R}$ there is $s_{0}\in\mathbb{R}$ with $0<s_{0}\le1$ such that $|H_{s}(y)-H(y)|<\varepsilon$ for every $s$ with $0<s\le s_{0}$.