On the stability test for reproducing kernel Hilbert spaces
This work provides a theoretical simplification for stability analysis in RKHSs, which is incremental but useful for researchers in functional analysis and control theory.
The paper tackles the problem of testing stability in reproducing kernel Hilbert spaces (RKHSs) by showing that the stability test can be reduced to studying kernel operators over test functions taking only ±1 values, generalizing BIBO stability results for linear time-invariant systems.
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.