  The recent results presented in arXiv:2202.05608 have led to significant
developments in achieving stable approximations of Helmholtz solutions by plane
wave superposition. The study shows that the numerical instability and
ill-conditioning inherent in plane wave-based Trefftz methods can be
effectively overcome with regularization techniques, provided there exist
accurate approximations in the form of expansions with bounded coefficients.
Whenever the target solution contains high Fourier modes, propagative plane
waves fail to yield stable approximations due to the exponential growth of the
expansion coefficients. Conversely, evanescent plane waves, whose modal content
covers high Fourier regimes, are able to provide both accurate and stable
results. The developed numerical approach, which involves constructing
evanescent plane wave approximation sets by sampling the parametric domain
according to a probability density function, results in substantial
improvements when compared to conventional propagative plane wave schemes. The
following work extends this research to the three-dimensional setting,
confirming the achieved results and introducing new ones. By generalizing the
3D Jacobi$-$Anger identity to complex-valued directions, we show that any
Helmholtz solution in a ball can be represented as a continuous superposition
of evanescent plane waves. This representation extends the classical Herglotz
one and provides a relevant stability result that cannot be achieved with the
use of propagative waves alone. The proposed numerical recipes have been
tailored for the 3D setting and extended with new sampling strategies involving
extremal systems of points. These methods are tested by numerical experiments,
showing the desired accuracy and bounded-coefficient stability, in line with
the two-dimensional case.
