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Motivated by the discrete dipole approximation (DDA) for the scattering ofelectromagnetic waves by a dielectric obstacle that can be considered as asimple discretization of a Lippmann-Schwinger style volume integral equationfor time-harmonic Maxwell equations, we analyze an analogous discretization ofconvolution operators with strongly singular kernels.  For a class of kernel functions that includes the finite Hilberttransformation in 1D and the principal part of the Maxwell volume integraloperator used for DDA in dimensions 2 and 3, we show that the method, whichdoes not fit into known frameworks of projection methods, can nevertheless beconsidered 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 convergeabsolutely. We use Ewald's method to obtain an exponentially fast convergentseries representation of this symbol and show that it is a bounded function,thereby allowing to describe the spectrum and the numerical range of thematrix.  It turns out that this numerical range includes the numerical range of theintegral operator, but that it is in some cases strictly larger. In these casesthe 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 wellposed, but the discretization method is unstable.

Stability Analysis of a Simple Discretization Method for a Class of Strongly Singular Integral Equations