Periodic PIC deposition shape and gather operators

On the torus and grid of Periodic uniform PIC grid and inner product, an algebraic assignment shape is any family of functions $w_j:{\mathbb{T}}_L^d\to [0,\mathrm{\infty })$ such that ${\sum }_{j\in J_N}{w}_j(x)=1$ for every $x$. For the smooth fixed-$\epsilon$ branch used in the force-error and ODE results, fix $\epsilon >0$, put ${\alpha }_{\epsilon }=e^{-\epsilon \mathrm{/}L}$ and choose the explicit even periodic kernel ${\eta }_{\epsilon }(x)=L^{-d}{\prod }_{a=1}^d\frac{1-{\alpha }_{\epsilon }^2}{1-2{\alpha }_{\epsilon }\cos (2\pi x_a\mathrm{/}L)+{\alpha }_{\epsilon }^2}\mathrm{.}$ For this smooth branch and each $x\in {\mathbb{T}}_L^d$, set $Z_{\epsilon ,h}(x)=h^d{\sum }_{j\in J_N}{\eta }_{\epsilon }(x_j-x),w_j^{\epsilon ,h}(x)=\frac{h^d{\eta }_{\epsilon }(x_j-x)}{Z_{\epsilon ,h}(x)}\mathrm{.}$ Each factor in the kernel denominator is positive, so $Z_{\epsilon ,h}(x)>0$ and the smooth-branch weights are well-defined. For any algebraic shape family above and finitely many particle positions $X=(X_1,\dots ,X_P)$ with charges $q_1,\dots ,q_P\in \mathbb{R}$, write $Q={\sum }_{p=1}^P{q}_p$ and define deposited charge density ${\rho }_j(X)=h^{-d}{\sum }_{p=1}^P{q}_p{w}_j(X_p)-Q\mathrm{/}{L}^d\mathrm{.}$ For any grid scalar $g$, its potential gather is $(G_{X}g{)}_p={\sum }_j{w}_j(X_p)g_j$. For any grid vector $E$, its field gather is defined componentwise by the same weights, $(G_{X}E{)}_p={\sum }_j{w}_j(X_p)E_j$. On the smooth fixed-$\epsilon$ branch, $w_j$ means the explicit family $w_j^{\epsilon ,h}$ above.