A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

Abstract

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 coordinateswhere 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 equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

Combined far‐ield operators in electromagnetic inverse scattering theory

“…This new Herglotz wave function and the kernel can now be used in place of V g and g in Theorem 2.4. The above approximation property was previously established by Ochs [16]. Here we present a di erent proof of this result which can also be extended to the case of electromagnetic waves (Section 3.2).…”

An Approximation Property of Importance in Inverse Scattering Theory

“…However, these results are only valid in R 2 and the techniques used cannot be extended to the three-dimensional case. The only corresponding approximation result in R 3 is due to Ochs [8], who needed to assume that D is star-like with respect to the origin. Since this last assumption is clearly not satisfactory from the point of view of inverse scattering, it is important to establish the denseness of Herglotz wave functions in H 1 (D) in a manner that is valid in both two and three dimensions, thus guaranteeing the validity of the linear sampling method in both of these cases.…”

Proper Generalized Decomposition applied to linear acoustic: A new tool for broad band calculation

“…Under some nonrestrictive geometric assumptions concerning Ω e , for the 2D case which we are considering, these functions can be represented by waves propagating in all possible 2D directions (see [21,22]…”