  We consider the multi-frequency inverse source problem for the scalar
Helmholtz equation in the plane. The goal is to reconstruct the source term in
the equation from measurements of the solution on a surface outside the support
of the source. We study the problem in a certain finite dimensional setting:
From measurements made at a finite set of frequencies we uniquely determine and
reconstruct sources in a subspace spanned by finitely many Fourier-Bessel
functions. Further, we obtain a constructive criterion for identifying a
minimal set of measurement frequencies sufficient for reconstruction, and under
an additional, mild assumption, the reconstruction method is shown to be
stable. Our analysis is based on a singular value decomposition of the
source-to-measurement forward operators and the distribution of positive zeros
of the Bessel functions of the first kind. The reconstruction method is
implemented numerically and our theoretical findings are supported by numerical
experiments.

Numerical Analysis