A version of Runge's theorem for the Helmholtz equation with applications to scattering theory
R.L. Ochs · Proceedings of the Edinburgh Mathematical Society · 1989
Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinates where C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equation in D can be approximated in the space by an entire Herglotz wave function with kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.