The sampling theorem plays an fundamental role for the recovery of continuous-time signals from discrete-time samples in the field of signal processing. The sampling theorem of non-band-limited signals has evolved into one of the most challenging problems. In this work, a generalized sampling theorem -- which builds on the Koopman operator -- is proved for signals in generator-bounded space (Theorem 1). It naturally extends the Nyquist-Shannon sampling theorem that, 1) for band-limited signals, the lower bounds of sampling frequency given by these two theorems are exactly the same; 2) the Koopman operator-based sampling theorem can also provide finite bound of sampling frequency for certain types of non-band-limited signals, which can not be addressed by Nyquist-Shannon sampling theorem. These types of non-band-limited signals include but not limited to, for example, inverse Laplace transform with limited imaginary interval of integration, and linear combinations of complex exponential functions. Moreover, the Koopman operator-based reconstruction algorithm is provided with theoretical result of convergence. By this algorithm, the sampling theorem is effectively illustrated on several signals related to sine, exponential and polynomial signals.

翻译：采样定理在信号处理领域对于从离散时间样本恢复连续时间信号起着基础性作用。非带限信号的采样定理已成为最具挑战性的问题之一。本文基于Koopman算子，针对生成元有界空间中的信号，证明了一个广义采样定理（定理1）。该定理自然拓展了奈奎斯特-香农采样定理，且具有以下性质：1）对于带限信号，两个定理给出的采样频率下界完全相同；2）基于Koopman算子的采样定理还能为某些类型的非带限信号提供有限的采样频率界，而这类信号无法通过奈奎斯特-香农采样定理处理。这些非带限信号类型包括但不限于：具有有限虚部积分区间的逆拉普拉斯变换、复指数函数的线性组合等。此外，本文还给出了基于Koopman算子的重构算法及其收敛性理论结果。通过该算法，本文有效展示了采样定理在正弦、指数及多项式等相关信号上的应用实例。