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 defined over the real line 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 , the space of essentially bounded (test) functions, and , the space of absolutely integrable functions. Considering all the Lebesgue measurable kernels in continuous-time and also the 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 . 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.

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再现核希尔伯特空间稳定性测试的改进

再生核希尔伯特空间 (RKHS) 是特殊的希尔伯特空间，其中所有评估函数都是线性且有界的。它们与称为核的正定映射一一对应。稳定的 RKHS 具有仅包含在实线上定义的函数并且绝对可积的附加属性。 RKHS 稳定性的充分必要条件在文献中是已知的：由核导出的积分算子必须被限制为 ，本质上有界（测试）函数的空间，和 ，绝对可积函数的空间之间的映射。考虑到连续时间和离散时间类中的所有勒贝格可测量核，我们表明稳定性测试可以简化为对测试函数的核算子的研究，该测试函数（几乎在所有地方）仅假设值 。它们代表了研究 RKHS 中任何单个元素的稳定性所需的相同功能。通过这种方式，RKHS 稳定性测试成为有关单个线性时不变系统的有界输入有界输出 (BIBO) 稳定性的简单结果的优雅概括。