标题： 基于过去导数样本的信号预测通过完美重构实现 显示英文标题

标题： Signal Prediction by Derivative Samples from the Past via Perfect Reconstruction

摘要： 本文研究了通过从位移不变空间中使用信号及其导数的非均匀样本进行完美重构来进行信号预测的问题。导数采样的主要优势在于能够降低采样率。我们基于周期性非均匀采样集推导出一种采样公式，该集合中包含位移不变空间中的导数。我们建立了该集合形成具有阶数$r-1$的完全插值序列（CIS）的充分必要条件。在此框架基础上，我们开发了一种在由紧支函数生成的位移不变空间中进行高效逼近的方法。以此为基础，我们提出了一种预测算法，该算法利用推导出的完美重构公式，根据有限数量的历史导数样本重建信号。最后，我们通过涉及三次样条和三阶Daubechies尺度函数的实际例子验证了我们的理论结果。 显示英文摘要

摘要： 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.