Scaled symmetric compactly supported kernel family

Fix an integer $d\ge 1$. An admissible smoothing kernel is a radial function $W\in C_c^2({\mathbb{R}}^d,[0,\mathrm{\infty }))$, supported in the closed Euclidean ball $\bar{B}(0,R)$ of some radius $R>0$ (Open ball, closed ball and sphere in a metric space), with ${\int }_{{\mathbb{R}}^d}W(z)dz=1$. Here radial means $W(z)=w(\mathrm{\mid }z\mathrm{\mid })$ for some function $w:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, and $C_c^2$ means twice continuously differentiable with compact support. All vector norms are the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and $\mathrm{\parallel }{D}^{2}W{\mathrm{\parallel }}_{\mathrm{\infty }}$ uses the induced operator norm. Throughout this pair, $L^{\mathrm{\infty }}$ norms are suprema of Euclidean norms, and norms of multilinear derivatives are the induced operator norms. Compactly supported integrals use The support of a function on ${\mathbb{R}}^n$ and its compactly supported Riemann integral, where $\mathrm{supp}f$ is the closure of the set on which $f$ is nonzero.

For $\epsilon >0$ define the scaled kernel $W_{\epsilon }(z)={\epsilon }^{-d}W(z\mathrm{/}\epsilon )$. Write $M_0=\mathrm{\parallel }W{\mathrm{\parallel }}_{\mathrm{\infty }}$, $M_1=\mathrm{\parallel }\mathrm{\nabla }W{\mathrm{\parallel }}_{\mathrm{\infty }}$, $M_2=\mathrm{\parallel }{D}^{2}W{\mathrm{\parallel }}_{\mathrm{\infty }}$, $m_2=\int \mathrm{\mid }z{\mathrm{\mid }}^{2}W(z)dz$, and $m_3=\int \mathrm{\mid }z{\mathrm{\mid }}^3\mathrm{\mid }\mathrm{\nabla }W(z)\mathrm{\mid }dz$. The norm functions and moment integrands are continuous with compact support. They are bounded and Riemann integrable on a bounding Jordan box by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set, so these constants are finite.

For continuous $f:{\mathbb{R}}^d\to \mathbb{R}$, define the full-space smoother and its kernel-gradient operator by $S_{\epsilon }f(x)={\int }_{{\mathbb{R}}^d}f(x-z)W_{\epsilon }(z)dz,D_{\epsilon }f(x)={\int }_{{\mathbb{R}}^d}f(y)\mathrm{\nabla }{W}_{\epsilon }(x-y)dy\mathrm{.}$ For each fixed $x$, both scalar integrands and every component of the vector integrand are continuous and compactly supported, hence Riemann integrable on a bounding Jordan box by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. Thus the displayed integrals exist as compactly supported Riemann integrals; the vector integral defining $D_{\epsilon }f$ is taken componentwise.

This page fixes $\epsilon$ in the finite-particle system. Limits $\epsilon \to 0$ are used only in the stated kernel-bias and joint-refinement results.