  Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces where
all the evaluation functionals are linear and bounded. They are in one-to-one
correspondence with positive definite maps called kernels. Stable RKHSs enjoy
the additional property of containing only functions and absolutely integrable.
Necessary and sufficient conditions for RKHS stability are known in the
literature: the integral operator induced by the kernel must be bounded as map
between $\mathcal{L}_{\infty}$, the space of essentially bounded (test)
functions, and $\mathcal{L}_1$, the space of absolutely integrable functions.
Considering Mercer (continuous) kernels in continuous-time and the entire
discrete-time class, we show that the stability test can be reduced to the
study of the kernel operator over test functions which assume (almost
everywhere) only the values $\pm 1$. They represent the same functions needed
to investigate stability of any single element in the RKHS. In this way, the
RKHS stability test becomes an elegant generalization of a straightforward
result concerning Bounded-Input Bounded-Output (BIBO) stability of a single
linear time-invariant system.
