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 coordinates
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091500006957/resource/name/S0013091500006957_eqnU1.gif?pub-status=live)
where 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 equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091500006957/resource/name/S0013091500006957_eqn1.gif?pub-status=live)
in D can be approximated in the space
by an entire Herglotz wave function
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091500006957/resource/name/S0013091500006957_eqnU2.gif?pub-status=live)
with kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.