Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and known nonlinear extensions of the fundamental lemma. Applying a transformation to the usual linear equation in Hankel matrices, we arrive at an alternative implicit kernel representation of the system trajectories while keeping the requirements on persistency of excitation. We show that this representation is equivalent to the solution of a specific kernel regression problem. We explore the possible structures of the underlying kernel as well as the system classes to which they correspond.

翻译：Willems等人的基本引理的推广与变体是近期研究的热点课题。本文探究并形式化了核回归与基本引理已知非线性扩展之间的联系。通过对汉克尔矩阵中常见的线性方程进行变换，我们在保持持续激励条件的同时，得到了系统轨迹的另一种隐式核表示。我们证明该表示等价于特定核回归问题的解，并进一步探讨了底层核的可能结构及其对应的系统类别。