  Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces in
one-to-one correspondence with positive definite maps called kernels. They are
widely employed in machine learning to reconstruct unknown functions from
sparse and noisy data. In the last two decades, a subclass known as stable
RKHSs has been also introduced in the setting of linear system identification.
Stable RKHSs contain only absolutely integrable impulse responses over the
positive real line. Hence, they can be adopted as hypothesis spaces to estimate
linear, time-invariant and BIBO stable dynamic systems from input-output data.
Necessary and sufficient conditions for RKHS stability are available in the
literature and it is known that kernel absolute integrability implies
stability. Working in discrete-time, in a recent work we have proved that this
latter condition is only sufficient. Working in continuous-time, it is the
purpose of this note to prove that the same result holds also for Mercer
kernels.
