The authors investigate a discrete approximation for the strongly singular integral equation \[ \lambda u(x) -\mbox{p.v.}\int_\Omega K(x-y)u(y) dy = f(x), \qquad x\in\Omega, \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^d\). The kernel \(K\) is defined by \[K(x) = p(x)/|x|^{d+2}, \] where \(p\) is a homogenous polynomial of degree \(2\), and \[\int_{S^{d-1}}p(x) ds = 0.\]
In the introduction the authors connect Equation (1) to the integral equation formulation of the Maxwell equation. The discrete dipole approximation (DDA) of (1) is given by \[ \lambda u_m - \sum_{x_n\in\Omega,n\neq m} h^d K(x_m-x_n)u_n = f(x_m),\qquad x_m\in \Omega,\tag{2} \] where \(h=1/N\) for a \(N\in\mathbb{N}\) and \(x_m=a_N+mh\), \(m\in\mathbb{Z}^d\), and an origin \(a_N\in\mathbb{R}^d\). The matrix of the DDA is denoted by \(T^N\) and is a finite section of the Toeplitz matrix \( T=(t_{mn})_{m,n\in\mathbb{Z}^d}\), with \(t_{m,n}=K(m-n)\) if \(m\neq n\), and \(0\) otherwise.
The authors prove that there is a compact convex set \(C\in\mathbb{C}\) such that (2) has a unique solution for \(\lambda\in \mathbb{C}\setminus C\) and \[ \|(\lambda\mathbb{I}-T^N)^{-1}\| \leq \mbox{dist}(\lambda,C)^{-1} .\]
Furthermore \(C\subset W(A)\), where \(W(A)\) is the numerical range of the convolution operator defined in (1). To prove this result the authors have to estimate the symbol of the Toeplitz operator \(T\) by using the method developed in [P. P. Ewald, Ann. der Phys. (4) 64, 253–287 (1921; JFM 48.0566.02)]. It is remarked that for \(\lambda\in C\setminus W(A)\) Equation (1) is well-posed but the discrete equation (2) is not. So, the DDA is unstable in this case.
In the final part of the paper several examples are presented for \(d=1,2,3\). For \(d=1\) the operator \(A\) is the discrete Hilbert transform with \(C=[-1,1]\). For \(d=2\) the examples are \(p(x)=x_1x_2|x|^{-4}\), \(p(x)=(x_1^2-x_2^2)|x|^{-4}\), and \(p(x)=(x_1+ix_2)^2|x|^{-4}\). Then the quasi-static Maxwell system is solved in dimension \(2\) and \(3\). Here the kernel \(K\) is matrix valued.