the field configurations received on So can be considered the "spatial waveforms" used by the communication system Current distribution on the Field distribution on radiating source the observation domain (2.1) wherein J(r) is the current distribution in the source domain D, E(r) is the field observed on the domain S2, and G is Green's function, i.e., the spatial impulse response. We suppose that E and J are square integrable functions. Standard L2 norm in Hilbert spaces is considered in the following. Since the energy of the currents is finite, the set of currents is bounded. A compact operator with bounded domain can be diagonalized using the Hilbert- Schmidt expansion [46, 63]. Let = , Uk (r'), ...} and u(r) = {u1 (r), u2(r), uk(r),...} be the left and the right singular basis of the Hilbert- Schmidt expansion. The current distribution on D and the field distribution on S can be expanded as: (2.2) M. D. Migliore E(r) =(r) = (2.3) wherein Xk and yk are the expansion coefficients of the currents and of the field distribution. As noted earlier, the energy of the current is limited. For the sake of simplicity, we will suppose ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| < 1. After the diagonalization of the radiation operator, the kth coefficient of the field is just a scaled version of the kth coefficient of the current: Yk=OkXk (2.4) wherein Ok is the k - th singular value of the radiation operator. The singular values are positive quantities sorted in descending way (01)02> ...) and Ok 0 when The current distribution J (r) can be represented by a point X having coordinates {x1, X2, Xk, ...} in an infinite dimensional Hilbert space (Fig. 2.4). Similarly, the field distribution E(r) can be represented by a point y, having coordinates {y1, y2, Yk, ...} in an infinite dimensional Hilbert space (Fig. 2.4). Accordingly, the radiation process maps points of the bounded spherical set X of all the possible current distributions on the source to points of the set Y