On the construction of problem-specific bases, frames, and wavelets in computational electromagnetics

The adequate analysis (transform) and synthesis (inverse transform) of functions (signals) in Hilbert space critically determine the overall performance of large-scale modeling and simulation of real-life problems in computational electromagnetics. Thereby, numerical schemes supporting the multiresolution and multiscale analysis lead to particularly well-structured and efficient formulations. This contribution is an attempt to further popularize some of the strikingly beautiful insights and enabling methodologies, which have been developing in the harmonic- and nonharmonic analysis community, among the scientists and engineers active in computational electromagnetics. The notions of orthogonal bases, nonorthogonal bases, dual bases, frames, and dual frames are briefly reviewed and presented in a unified fashion. It is shown that the spectral-domain far-field asymptotic tails of Green´s functions associated with electromagnetic wave propagation problems can be utilized to create finite-support orthogonal wavelets, satisfying the axioms of the multiresolution analysis.