On the stability of reproducing kernel Hilbert spaces of discrete-time impulse responses

Reproducing kernel Hilbert spaces (RKHSs) have proved themselves to be key tools for the development of powerful machine learning algorithms, the so-called regularized kernel-based approaches.Recently, they have also inspired the design of new linear system identification techniques able to challenge classical parametric prediction error methods. These facts motivate the study of the RKHS theory within the control community. In this note, we focus on the characterization of stable RKHSs, i.e. RKHSs of functions representing stable impulse responses. Related to this, working in an abstract functional analysis framework, Carmeli et al. (2006) has provided conditions for an RKHS to be contained in the classical Lebesgue spaces ℒp. In particular, we specialize this analysis to the discrete-time case with p=1. The necessary and sufficient conditions for the stability of an RKHS are worked out by a quite simple proof, more easily accessible to the control community.