Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann-Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular kernels. For a class of kernel functions that includes the finite Hilbert transformation in 1D and the principal part of the Maxwell volume integral operator used for DDA in dimensions 2 and 3, we show that the method, which does not fit into known frameworks of projection methods, can nevertheless be considered as a finite section method for an infinite block Toeplitz matrix. The symbol of this matrix is given by a Fourier series that does not converge absolutely. We use Ewald's method to obtain an exponentially fast convergent series representation of this symbol and show that it is a bounded function, thereby allowing to describe the spectrum and the numerical range of the matrix. It turns out that this numerical range includes the numerical range of the integral operator, but that it is in some cases strictly larger. In these cases the discretization method does not provide a spectrally correct approximation, and while it is stable for a large range of the spectral parameter $\lambda$, there are values of $\lambda$ for which the singular integral equation is well posed, but the discretization method is unstable.

翻译：受电磁波被介电障碍物散射的离散偶极子近似（DDA）启发（该近似可视为时谐麦克斯韦方程组中Lippmann-Schwinger型体积积分方程的简单离散化），我们分析了一类具有强奇异核的卷积算子的类似离散化方法。对于一类核函数（包括一维有限希尔伯特变换，以及二、三维DDA中使用的麦克斯韦体积积分算子的主部），我们证明该方法虽不适用于已知的投影方法框架，但仍可被视为无限块Toeplitz矩阵的有限截面方法。该矩阵的符号由非绝对收敛的傅里叶级数给出。我们利用埃瓦尔德方法获得该符号的指数快速收敛级数表示，并证明其为有界函数，从而得以描述该矩阵的谱和数值范围。结果表明，该数值范围包含积分算子的数值范围，但在某些情况下严格更大。在这些情况下，离散化方法无法提供谱正确的近似；虽然该方法在谱参数$\lambda$的大范围内稳定，但存在某些$\lambda$值使得奇异积分方程适定，而离散化方法却不稳定。