On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces

Abstract

SUMMARYLet D ⊂ R 3 be a bounded domain with connected boundary 9D of class C 2 . It is shown that Herglotz wave functions are dense in the space of solutions to the Helmholtz equation with respect to the norm in H 1 (D) and that the electric ÿelds of electromagnetic Herglotz pairs are dense in the space of solutions to curl curl E = k 2 E with respect to the norm in H curl (D). Two proofs are given in each case, one based on the denseness of the traces of Herglotz wave functions on 9D and the other on variational methods.

A new non-iterative singular sources method for the reconstruction of piecewise constant media.

“…We observe that for the method of singular sources we only need denseness in the set of solutions to the inhomogeneous Helmholtz equation. This denseness can be proven without the restriction on the the domains G under consideration, see for example [6]. The following results generalize Theorem 2.6 of [6] to the case of an inhomogeneous background medium.…”

Combined far‐ield operators in electromagnetic inverse scattering theory

“…Let E int be the interior electric ÿeld that solves (TPM) with boundary data h := (1=ik) × ∇ × E e . Then from the results of Reference [9] (for comments on nonsmooth boundaries see Reference [2]) for each ¿0 we can ÿnd an electromagnetic Herglotz pair with kernel g (·; z) ∈ L 2 t ( ) such that E g approximates E int ∈ M(D) with respect to the H (curl; D) norm, and moreover E g is an approximate solution to (42), i.e. where arbitrary small ¿0 measures the approximation by the Herglotz function, the arbitrary small ¿0 measures the perturbation of (42) to ensure a right-hand side in the range of D and ¿0 is the regularization parameter corresponding to which additionally satisÿes → 0 as → 0.…”

“…Then from References [9,10] (see also Reference [15]) for every ¿0 we can ÿnd a g (·; z) such that the corresponding Herglotz wave function V g (·; z) satisÿes…”

“…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.…”

Approximation by herglotz wave functions

“…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).…”