We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.

翻译：本文提出了一种基于Toeplitz结构的框架，用于数据驱动的动力系统中线性演化算子的谱估计。该方法聚焦于从平衡轨迹中提取转移算子和Koopman算子（无需知晓底层运动方程），通过对无穷小生成元施加Toeplitz滤波器来提取特征值、特征函数及谱测度。通过设计可融入自伴性或斜对称性等结构先验知识。该方法具有统计一致性且计算高效，利用了统计学习中常用的原始算法与对偶算法。在确定性与混沌系统上的数值实验表明，该框架能够恢复传统数据驱动方法无法触及的谱特性。