Cell-volume product integration for the scattering kernel

Assume Countable Choice. Let $D,q,k$ be as in The three-dimensional volume integral scattering model, let $f\in L^2(D;\mathbb{C})$, and assume $0<\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$. Let $P_N$ be the dyadic cell-average projection of Dyadic cell-average projections converge strongly in L2 of a bounded set, and put $K_N:=P_N{K}_k$ and $f_N:=P_{N}f$. This is a finite-rank volume product-integration rule on $L^2(D)$; the projection and kernel convention are fixed by those definitions.

For each positive-measure cell $E_{N,j}$ set ${\varphi }_{N,j}:=\mathrm{\mid }{E}_{N,j}{\mathrm{\mid }}^{-1\mathrm{/}2}{\mathbf{1}}_{E_{N,j}}$. The exact product-cell weights and right-hand side are $A_{ij}^{(N)}:=⟨K_k{\varphi }_{N,j},{\varphi }_{N,i}⟩=\frac{1}{\sqrt{\mathrm{\mid }{E}_{N,i}\mathrm{\mid }\mathrm{\mid }{E}_{N,j}\mathrm{\mid }}}{\int }_{E_{N,i}}{\int }_{E_{N,j}}{G}_k(x-y)q(y)dydx,b_i^{(N)}:=\frac{1}{\sqrt{\mathrm{\mid }{E}_{N,i}\mathrm{\mid }}}{\int }_{E_{N,i}}f(x)dx\mathrm{.}$ For each cell pair, the double integral is absolutely convergent: the kernel $G_k(x-y)q(y)$ is in $L^2(D\times D)$ and ${\varphi }_{N,i}(x){\varphi }_{N,j}(y)$ has product $L^2$ norm one, so Cauchy--Schwarz makes their product integrable. Completed-product Fubini under Countable Choice then gives the displayed equality with the inner product. The finite system is $(I-A^{(N)})c=b^{(N)}$, and its reconstruction is $u_N={\sum }_j{c}_j{\varphi }_{N,j}$. Equivalently, it solves $(I-K_N)u_N=f_N$ in $V_N$. Its far-field output is the exact cell integral ${\mathcal{F}}_N(c)(\theta ):={\mathcal{F}}_k[u_N](\theta )={\sum }_j\frac{c_j}{4\pi \sqrt{\mathrm{\mid }{E}_{N,j}\mathrm{\mid }}}{\int }_{E_{N,j}}{e}^{-ik\theta \cdot y}q(y)dy\mathrm{.}$ A singular finite matrix has status singular; the definition makes no unconditional invertibility claim. The double integrals are taken over whole cell pairs, including diagonal pairs; no point value of $G_k$ at its singular diagonal is sampled. Finiteness is established by The volume integral operator is well defined and bounded on L2 and The volume far-field map is bounded and is the uniform far-field limit.

The rule uses exact cell weights as a mathematical discretization. This definition makes no claim about floating-point evaluation or approximate cell weights.