An approximation property of importance in inverse scattering theory (Q2758141)

The authors give a new proof of the fact that every \(H^1\)-solution of the Helmholtz equation in a bounded region \(D\subset\mathbb{R}^2\) (the complement of which is connected) can be approximated arbitrarily well in the \(H^1\)-norm by Herglotz wave functions, i.e by function of the form NEWLINE\[NEWLINEv (x)= \int_\Omega g(y) \exp(ikx\cdot y)ds(y),\quad g\in L^2(\Omega),NEWLINE\]NEWLINE where \(\Omega\) denotes the unit circle in \(\mathbb{R}^2\). As indicated in this paper, this approximation property is, e.g., of essential importance for the linear sampling theory in inverse scattering.