This paper investigates signal prediction through the perfect reconstruction of signals from shift-invariant spaces using nonuniform samples of both the signal and its derivatives. The key advantage of derivative sampling is its ability to reduce the sampling rate. We derive a sampling formula based on periodic nonuniform sampling (PNS) sets with derivatives in a shift-invariant space. We establish the necessary and sufficient conditions for such a set to form a complete interpolating sequence (CIS) of order $r-1$. This framework is then used to develop an efficient approximation scheme in a shift-invariant space generated by a compactly supported function. Building on this, we propose a prediction algorithm that reconstructs a signal from a finite number of past derivative samples using the derived perfect reconstruction formula. Finally, we validate our theoretical results through practical examples involving cubic splines and the Daubechies scaling function of order 3.

翻译：本文研究通过利用信号及其导数的非均匀采样实现平移不变空间中信号的完美重构，进而进行信号预测。导数采样的主要优势在于能够降低采样率。我们推导了基于平移不变空间中周期性非均匀采样（PNS）集及其导数的采样公式。我们建立了此类采样集构成$r-1$阶完全插值序列（CIS）的充分必要条件。该框架随后被用于在由紧支撑函数生成的平移不变空间中构建高效逼近方案。在此基础上，我们提出一种预测算法，该算法利用推导出的完美重构公式，通过有限个历史导数样本来重构信号。最后，我们通过三次样条函数和三阶Daubechies尺度函数的实际算例验证了理论结果。