Summary: The problem of the identification of the electromagnetic source which produces an assigned radiation pattern is ill-posed: the solution is, in general, not unique and it does not depend continuously on the data. In this paper, we treat in detail these two aspects of the problem. First of all, we reconsider the radiation problem in the very general setting of the Sobolev spaces in order to make more acceptable, from a physical viewpoint, the conditions which have to be imposed on the electromagnetic sources. Then by the use of the Euclidean character of the Hilbert spaces, we decompose the sources into a radiating and a non-radiating component. We determine the subspace of the radiating sources and we find the basis spanning this subspace.
Particular attention is then devoted to the case of the linear antenna. In this case, the solution of the problem is unique but it does not depend continuously on the data. We may, however, implement the problem taking into account a bound on the Ohmic losses. This is sufficient to restore the continuity. Finally, a method of variational regularization (in the sense of Tikhonov) is discussed in detail.