An Approximation Property of Importance in Inverse Scattering Theory

Abstract

A key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle D from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in D can be approximated in H 1 (D) by Herglotz wave functions.To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space H 1 (D) norm.

Approximation by herglotz wave functions

“…Second by an approximation of the domain 'from outside' the approximation is made 'global' (Theorem 2) which in turn makes it possible to approximate boundary data (Theorems 3 and 4). Essentially, we shall be able to reproduce the results of Reference [3] and generalize them a bit. Furthermore, our argument carries over to other equations for which inverse scattering problems exist (Maxwell's equation, linear isotropic elasticity, etc.).…”

“…The argument for the ÿrst part has been given in Reference [3] (but has already been used in Reference [7]). The second part follows from regularity theory.…”

“…We may refer to the survey article [1] where it is described in detail why it is crucial for this method to be able to approximate solutions of the reduced wave equation by 'Herglotz wave functions' to be described below. So approximation problems of this kind have been the object of several studies (see References [2][3][4] and the literature cited there).…”

Combined far‐ield operators in electromagnetic inverse scattering theory

“…However, the analysis is not as straightforward as might at ÿrst be expected. In particular, it is now necessary to rely on new approximation properties of Herglotz wave functions and electromagnetic Herglotz pairs as developed in References [9,10] as well as requiring a factorization of the modiÿed far-ÿeld operator, neither of which was needed in References [7,8]. We note that the approach used here to eliminate the problem of eigenvalues in the linear sampling method relies heavily on the polarization of the incident and scattered ÿelds and hence is only applicable to the case of electromagnetic scattering.…”

“…Therefore by the bounded inverse theorem S −1 is continuous. In terms of the operator B, using (10) and (11), we can now rewrite (7) as…”