  Reproducing kernel Hilbert spaces (RKHSs) are key spaces for machine learning
that are becoming popular also for linear system identification. In particular,
the so-called stable RKHSs can be used to model absolutely summable impulse
responses. In combination e.g. with regularized least squares they can then be
used to reconstruct dynamic systems from input-output data. In this paper we
provide new structural properties of stable RKHSs. The relation between stable
kernels and other fundamental classes, like those containing absolutely
summable or finite-trace kernels, is elucidated. These insights are then
brought into the feature space context. First, it is proved that any stable
kernel admits feature maps induced by a basis of orthogonal eigenvectors in l2.
The exact connection with classical system identification approaches that
exploit such kind of functions to model impulse responses is also provided.
Then, the necessary and sufficient stability condition for RKHSs designed by
formulating kernel eigenvectors and eigenvalues is obtained. Overall, our new
results provide novel mathematical foundations of stable RKHSs with impact on
stability tests, impulse responses modeling and computational efficiency of
regularized schemes for linear system identification.
