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.
AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25
