Best bells for the scattering by perfectly conducting cylinders

Application of the local bell-modulated cosine bases of Coifman and Meyer to the problem of scattering by PEC cylinders leads to a sparse Method of Moments interaction matrix. To optimize the sparsity representation it is imperative to find bell functions with nice Fourier radiation properties. Here we pose the problem of bell optimization in terms of Chebyshev's inequality, resulting in a family of elliptic bells (depending on elliptic functions) parametrized by a real parameter. The effect of finite curvature being, as a first approximation, a deformation of the originally pythagorean bell into a non-pythagorean bell, we then extend the formulation to include general non-pythagorean bells. We propose a class of Kaiser-Bessel bells, well-known from signal and image processing theory for their nice radiation properties. We implement the method to scattering by elliptic and cardioid-like cylinders to illustrate the technique. Based on various specific indicators measuring sparsity, fastness and accuracy, we find that a pertinent Kaiser-Bessel bell performs generally better than the elliptic bells.