Minimum source regions of far fields as probable bounds of support

It is well known that, for a given far field radiation pattern, there is a minimum source region V_min such that in order for the field to be produced by a source of support V, then V_min ⊆ V. There are classical and modern techniques for the estimation of bounding regions B_min such that V_min ⊆ B_min. This paper investigates a fundamental open question in the electromagnetic theory of minimum source regions which has much practical importance for inverse source and scattering problems. In particular, clearly one can obtain estimates of the minimum source region; but how much information do they provide about the true support of a given unknown source whose far fields are measured? It is shown that, for a rather general class of sources of support V, the respective minimum source region and tight estimates of this region obtained from far field data are very likely to coincide with or at least to be close to the original support V (within bounds estimated in the paper). Thus, in inverse source and scattering problems, estimates of these minimum source regions, obtained from far field data, are probabilistically tight estimates of the sought-after support.