  Extreme mass-ratio inspirals (EMRIs) are expected to have considerable
eccentricity when emitting gravitational waves (GWs) in the LISA band.
Developing GW templates that remain phase accurate over these long inspirals
requires the use of second-order self-force theory and practical second-order
self-force calculations are now emerging for quasi-circular EMRIs. These
calculations rely on effective-source regularization techniques in the
frequency domain that presently are specialized to circular orbits. Here we
make a first step towards more generic second-order calculations by extending
the frequency domain effective-source approach to eccentric orbits. In order to
overcome the slow convergence of the Fourier sum over radial modes, we develop
a new extended effective-sources approach which builds upon the method of
extended particular solutions. To demonstrate our new computational technique
we apply it a toy scalar-field problem which is conceptually similar to the
gravitational case.
