Periodic cardinal-spline assignment and unitary mesh transform

Fix neutral point-source data $(A,V,\mathrm{\Lambda },{\kappa }_k,(x_i,q_i{)}_{i=1}^n)$ from Neutral point sources and Fourier data on a three-dimensional torus, an Ewald parameter $\alpha >0$, integers $M_1,M_2,M_3\ge 1$, an assignment order $P\ge 2$, and write $N_g=M_1{M}_2{M}_3$. Let $\mathcal{G}={\prod }_{r=1}^3\{0,\dots ,M_r-1\}$ and take $W_P$ from Causal cardinal spline window. Define the periodic causal assignment weights, for $t\in [0,1{)}^3$ and $\mathrm{\ell }\in \mathcal{G}$, by $w_{\mathrm{\ell }}(t)={\prod }_{r=1}^3{\sum }_{j\in \mathbb{Z}}{W}_P(M_r{t}_r-{\mathrm{\ell }}_r-j{M}_r)\mathrm{.}$ The sums are finite because $W_P$ is supported on $[0,P]$. They collect every support index into its grid residue; if $P>M_r$, distinct support indices that wrap to the same residue are added. The spread grid is $g_{\mathrm{\ell }}={\sum }_{i=1}^n{q}_i{w}_{\mathrm{\ell }}(t_i),t_i=A^{-1}{x}_i(\mathrm{m}\mathrm{o}\mathrm{d}{\mathbb{Z}}^3)\mathrm{.}$

Use the negative-sign unitary tensor DFT convention ${\widehat{g}}_M(k)=N_g^{-1\mathrm{/}2}{\sum }_{\mathrm{\ell }\in \mathcal{G}}{g}_{\mathrm{\ell }}{e}^{-2\pi i{\sum }_{r=1}^3{k}_r{\mathrm{\ell }}_r\mathrm{/}{M}_r},$ which is the product transform of The unitary discrete Fourier transform on $\mathbb{Z}\mathrm{/}N\mathbb{Z}$. Signed frequency representatives are chosen before evaluating windows. A retained set $\mathcal{K}\subset {\mathbb{Z}}^3\setminus \{0\}$ is finite, symmetric under $k\mapsto -k$, injective modulo $(M_1,M_2,M_3)$, and satisfies $\mathrm{\mid }{k}_r\mathrm{\mid }<M_r\mathrm{/}2$ for each coordinate. In particular, even-grid Nyquist modes are not silently retained.

For $u\in \mathbb{R}$, put $\mathrm{sinc}(u)=\sin (u)\mathrm{/}u$ when $u\mathrm{\ne }0$ and $\mathrm{sinc}(0)=1$. Define $D(k)={\prod }_{r=1}^3{e}^{-\pi iP{k}_r\mathrm{/}{M}_r}\mathrm{sinc}(\pi k_r\mathrm{/}{M}_r{)}^P\mathrm{.}$ Only modes with $D(k)\mathrm{\ne }0$ are admissible for division. A computed implementation must first certify a strictly positive lower bound for $\mathrm{\mid }D(k)\mathrm{\mid }$; otherwise it returns denominator-inconclusive without dividing. For $k\in \mathcal{K}$, define $a_k(t)=e^{-2\pi ik\cdot t},b_k(t)=\frac{{\sum }_{\mathrm{\ell }\in \mathcal{G}}{w}_{\mathrm{\ell }}(t)e^{-2\pi i{\sum }_r{k}_r{\mathrm{\ell }}_r\mathrm{/}{M}_r}}{\overline{D(k)}},{\rho }_k={\sum }_i{q}_i{a}_k(t_i),z_k=\frac{\sqrt{N_g}{\widehat{g}}_M(k)}{\overline{D(k)}}\mathrm{.}$ The denominator is the complex conjugate of the printed window multiplier; this sign is part of the definition. Set $c_k=\frac{4\pi }{V\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2}{e}^{-\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2\mathrm{/}(4{\alpha }^2)}\mathrm{.}$ The exact retained reciprocal energy is $E_{\mathcal{K}}=\frac{1}{2}{\sum }_{k\in \mathcal{K}}{c}_k\mathrm{\mid }{\rho }_k{\mathrm{\mid }}^2,$ and the mesh reciprocal energy is $E_M=\frac{1}{2}{\sum }_{k\in \mathcal{K}}{c}_k\mathrm{\mid }{z}_k{\mathrm{\mid }}^2\mathrm{.}$

Let ${\mathfrak{d}}_k(t)$ be the declared physical mesh derivative: it is ${\mathrm{\nabla }}_x{b}_k(t)=A^{-T}{\mathrm{\nabla }}_t{b}_k(t)$ wherever the classical derivative exists. A grid knot is a point with $M_r{t}_r\in \mathbb{Z}$ for some coordinate $r$. At a $P=2$ knot, take the one-sided vector limits from all adjacent open mesh boxes. Define ${\mathfrak{d}}_k(t)$ to be any selected one-sided limit or the arithmetic average of all those limits. This is a procedure value and does not assert that the classical derivative exists there. The energy-gradient mesh force is $F_{M,i}=-q_i{\sum }_{k\in \mathcal{K}}{c}_k\mathrm{Re}\text{\hspace{0.17em}⁣}(\overline{z_k}{\mathfrak{d}}_k(t_i))\mathrm{.}$ Define the diagonal mesh-self correction and corrected reciprocal energy by $C_M=\frac{1}{2}{\sum }_i{q}_i^2{\sum }_{k\in \mathcal{K}}{c}_k(\mathrm{\mid }{b}_k(t_i){\mathrm{\mid }}^2-1),E_M^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}=E_M-C_M\mathrm{.}$ Whether the declared force is the derivative of this corrected energy, and the error bound it satisfies, are proved separately.

• ${\mathrm{\nabla }}_x=A^{-T}{\mathrm{\nabla }}_t$ is the physical derivative in the cell coordinates $x=At$.
• Every reciprocal mode omitted from $\mathcal{K}$, including Nyquist modes, belongs to the separately bounded reciprocal truncation error.
• This definition makes no accuracy claim about an FFT or a floating-point implementation. The error theorem supplies those conditions separately.