The Whitney embedding theorem is a basic result of differential topology and it is natural to ask for versions of the embedding theorem for time series data. Embedding theorems for dynamical time series data were claimed by Takens (1981) and proved in a different setting by Sauer, Yorke, and Casdagli (1991). Those results were stated assuming the observation function to be parametrized. A possibly more pertinent setting is to assume the dynamical system to be parametrized, with the observation function fixed, for example, as a projection to a certain coordinate. We prove an embedding theorem in such a setting. Our proof introduces a technique that relies on the notion of Lebesgue points.

惠特尼嵌入定理是差分拓扑的基本结果，很自然地需要时间序列数据的嵌入定理版本。Takens（1981）提出了动态时间序列数据的嵌入定理，Sauer，Yorke和Casdagli（1991）在不同的背景下证明了该定理。陈述这些结果时假设观察函数已参数化。可能更相关的设置是假设要对动力学系统进行参数化，并固定观察功能，例如将其投影到某个坐标。我们证明了在这种情况下的嵌入定理。我们的证明引入了一种依赖于Lebesgue点的概念的技术。