Uni-variable cross-spectral densities

We introduce a novel class of planar, scalar sources whose cross-spectral density is a function of a single complex variable. For a typical pair of source points, the modulus and argument of such a variable equal the product of their radial coordinates and the difference of their angular coordinates, respectively. As functions of a single complex variable, the corresponding cross-spectral densities (CSD) are expandable in power series in their convergence domain, and a virtually infinite number of source models can be devised. All such sources are shown to have vortex fields as modes. The closed analytical form of these uni-variable CSDs makes it possible to evaluate in a simple way the significant quantities characterizing the sources and the ones of the fields they radiate. The basic tool leading to the definition of the uni-variable CSDs are the reproducing-kernel Hilbert spaces. As an example, the CSD derived from the so called Szeg & ouml; kernel is studied in some detail, and its features are derived, together with of those of the radiated field in the far zone.