The set of the regular and radiating spherical vector wave functions (SVWF) is shown to be complete in L2(S) where S is a Lyapunov surface. The completeness fails when k2 is an eigenvalue of the Dirichlet problem for the Helmholtz equation (▽2 + k2)F = 0 in the region Di bounded by S. On the other hand, the set of radiating SVWF is shown to be complete for all values of k2. It is also proved that any vector function, which is continuous in Di + S and satisfies the Helmholtz equation in Di., can be approximated uniformly in Di and in the mean square sense on S by a sequence of linear combinations of the regular SVWF (assuming the set is complete). Similar results are obtained for the exterior problem with the set of radiating SVWF. These results are extended to the set composed of the regular and radiating SVWF on two nonintersecting Lyapunov surfaces, one of which encloses the other.
All Science Journal Classification (ASJC) codes
- Applied Mathematics