Inverse source problem and minimum-energy sources

We present a new linear inversion formalism for the scalar inverse source problem in three-dimensional and one-dimensional (1D) spaces, from which a number of previously unknown results on minimum-energy (ME) sources and their fields readily follow. ME sources, of specified support, are shown to obey a homogeneous Helmholtz equation in the interior of that support. As a consequence of that result, the fields produced by ME sources are shown to obey an iterated homogeneous Helmholtz equation. By solving the latter equation, we arrive at a new Green-function representation of the field produced by a ME source. It is also shown that any square-integrable (L2), compactly supported source that possesses a continuous normal derivative on the boundary of its support must possess a nonradiating (NR) component. A procedure based on our results on the inverse source problem and ME sources is described to uniquely decompose an L2 source of specified support and its field into the sum of a radiating and a NR part. The general theory that is developed is illustrated for the special cases of a homogeneous source in 1D space and a spherically symmetric source.