  The notion of reproducing kernel Hilbert space (RKHS) has emerged in system
identification during the past decade. In the resulting framework, the impulse
response estimation problem is formulated as a regularized optimization defined
on an infinite-dimensional RKHS consisting of stable impulse responses. The
consequent estimation problem is well-defined under the central assumption that
the convolution operators restricted to the RKHS are continuous linear
functionals. Moreover, according to this assumption, the representer theorem
hold, and therefore, the impulse response can be estimated by solving a
finite-dimensional program. Thus, the continuity feature plays a significant
role in kernel-based system identification. This paper shows that this central
assumption is guaranteed to be satisfied in considerably general situations,
namely when the kernel is an integrable function and the input signal is
bounded. Furthermore, the strong convexity of the optimization problem and the
continuity property of the convolution operators imply that the kernel-based
system identification admits a unique solution. Consequently, it follows that
kernel-based system identification is a well-defined approach.
