Kernel absolute summability is only sufficient for RKHS stability
This resolves a theoretical gap in RKHS stability for linear dynamic systems, but is incremental as it clarifies an existing condition rather than introducing new methods.
The paper addresses the open question of whether absolute summability of a kernel is equivalent to stability in Reproducing Kernel Hilbert Spaces (RKHSs) for linear system identification, proving that this correspondence does not hold by providing a counterexample of stable RKHSs induced by non-absolutely summable kernels.
Regularized approaches have been successfully applied to linear system identification in recent years. Many of them model unknown impulse responses exploiting the so called Reproducing Kernel Hilbert spaces (RKHSs) that enjoy the notable property of being in one-to-one correspondence with the class of positive semidefinite kernels. The necessary and sufficient condition for a RKHS to be stable, i.e. to contain only BIBO stable linear dynamic systems, has been known in the literature at least since 2006. However, an open question still persists and concerns the equivalence of such condition with the absolute summability of the kernel. This paper provides a definite answer to this matter by proving that such correspondence does not hold. A counterexample is introduced that illustrates the existence of stable RKHSs that are induced by non-absolutely summable kernels.