Let the pdf of a standard multivariate normal distribution be \begin{equation} p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2). \end{equation} Consider a continuous function $f(\mathbf{s})$ supported on a compact set $\mathcal{S}$. Then let \begin{equation} I_{\sigma}(\mathbf{s})=\int_{\mathbb{R}^{k}} p_{Z}(\mathbf{z}-\mathbf{s}) f(\mathbf{z}) d\Omega, \mathbf{s}\in \mathcal{S}. \end{equation} My question is whether $I_{\sigma}(\mathbf{s})$ converges to $f(\mathbf{s})$ uniformly on $\mathcal{S}$ as $\sigma \to 0^{+}$. Can the condition of the compactness of $\mathcal{S}$ be relaxed? In addition, tight nonasymptotic lower and upper bounds on $I_{\sigma}(\mathbf{s})$ are more welcome.

Let the pdf of a multivariate normal distribution be \begin{equation} p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2). \end{equation} Consider a continuous function $f(\mathbf{s})$ supported on a compact set $\mathcal{S}$. Then let \begin{equation} I_{\sigma}(\mathbf{s})=\int_{\mathbb{R}^{k}} p_{Z}(\mathbf{z}-\mathbf{s}) f(\mathbf{z}) d\Omega, \mathbf{s}\in \mathcal{S}. \end{equation} My question is whether $I_{\sigma}(\mathbf{s})$ converges to $f(\mathbf{s})$ uniformly on $\mathcal{S}$ as $\sigma \to 0^{+}$. Can the condition of the compactness of $\mathcal{S}$ be relaxed? In addition, tight nonasymptotic lower and upper bounds on $I_{\sigma}(\mathbf{s})$ are more welcome.

Let the pdf of a standard multivariate normal distribution be \begin{equation} p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2). \end{equation} Consider a continuous function $f(\mathbf{s})$ supported on a compact set $\mathcal{S}$. Then let \begin{equation} I_{\sigma}(\mathbf{s})=\int_{\mathbb{R}^{k}} p_{Z}(\mathbf{z}-\mathbf{s}) f(\mathbf{z}) d\Omega, \mathbf{s}\in \mathcal{S}. \end{equation} My question is whether $I_{\sigma}(\mathbf{s})$ converges to $f(\mathbf{s})$ uniformly on $\mathcal{S}$ as $\sigma \to 0^{+}$. Can the condition of the compactness of $\mathcal{S}$ be relaxed? In addition, nonasymptotic lower and upper bounds on $I_{\sigma}(\mathbf{s})$ is welcome.

Let the pdf of a standard multivariate normal distribution be \begin{equation} p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2). \end{equation} Consider a continuous function $f(\mathbf{s})$ supported on a compact set $\mathcal{S}$. Then let \begin{equation} I_{\sigma}(\mathbf{s})=\int_{\mathbb{R}^{k}} p_{Z}(\mathbf{z}-\mathbf{s}) f(\mathbf{z}) d\Omega, \mathbf{s}\in \mathcal{S}. \end{equation} My question is whether $I_{\sigma}(\mathbf{s})$ converges to $f(\mathbf{s})$ uniformly on $\mathcal{S}$ as $\sigma \to 0^{+}$. Can the condition of the compactness of $\mathcal{S}$ be relaxed? In addition, tight nonasymptotic lower and upper bounds on $I_{\sigma}(\mathbf{s})$ are more welcome.