A scheme to generalize signal theory and its application to electromagnetics

The scheme summarized in this work is being applied to generalize the solution to problems described by differential operators with some boundary conditions applying over signals spaces which depend on several dimensions -for instance, tau equiv r-. These analyses directly connect with Green´s function theory and its spectral representation, a fundamental tool in the formulation and resolution of physical problems (the author is particularly involved in the analysis of EM problems). Another important extension of this generalized scheme is concerned with the possibility of its application to generalize the study to complex variable signal spaces -with tau equiv z = x + jy-. The practical application of these results will be the generalization of the analysis of radiation and scattering problems in EM when time and/or space coordinates are continued into complex ones -complex signal theory [E. Gago-Ribas et al., 2003]. This problem was the initial aim to try to obtain the general scheme in real variable summarized in the present work and which is currently used also to present the signals and systems theory to undergraduate and postgraduate students.