Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces in one-to-one correspondence with positive definite maps called kernels. They are widely employed in machine learning to reconstruct unknown functions from sparse and noisy data. In the last two decades, a subclass known as stable RKHSs has been also introduced in the setting of linear system identification. Stable RKHSs contain only absolutely integrable impulse responses over the positive real line. Hence, they can be adopted as hypothesis spaces to estimate linear, time-invariant and BIBO stable dynamic systems from input-output data. Necessary and sufficient conditions for RKHS stability are available in the literature and it is known that kernel absolute integrability implies stability. Working in discrete-time, in a recent work we have proved that this latter condition is only sufficient. Working in continuous-time, it is the purpose of this note to prove that the same result holds also for Mercer kernels.

翻译：再生核希尔伯特空间是与称为核的正定映射一一对应的特殊希尔伯特空间。它们被广泛用于机器学习中，从稀疏含噪数据中重建未知函数。在过去二十年中，在线性系统辨识的背景下，还引入了一类称为稳定RKHS的子类。稳定RKHS仅包含正实轴上的绝对可积脉冲响应。因此，它们可以作为假设空间，从输入输出数据估计线性时不变且BIBO稳定的动态系统。文献中已有关于RKHS稳定性的充要条件，并且已知核的绝对可积性蕴含稳定性。在离散时间框架下，我们最近的工作已证明后者仅仅是充分条件。在连续时间框架下，本工作旨在证明这一结论对Mercer核同样成立。