再生核空间中的随机采样与重构算法

采样理论研究如何从采样点上的采样值重构或逼近原始信号。随机采样是指采样点从定义在有界区域上的概率分布中独立随机选取，采样值视采样设备的特点和性能而定，可能是在采样点的信号值、在采样点附近信号的某种加权平均值等。本项目研究再生核空间中的随机采样与重构问题，可概括为三个方面：（1）对不同类型的采样值，分析随机采样方法的稳定性；（2）提出相应的重构算法，使得可从有限个随机采样值重构或逼近原始信号，并分析重构算法的收敛速度、稳定性以及采样量与取得相应重构误差的概率之间的关系；（3）对信号进行统计推断，包括对原始信号的假设检验和对重构信号的统计性质分析。本项目综合利用信号分析、逼近论和统计推断等方面的知识对上述问题做出研究。这些研究将会丰富随机采样的数学理论，同时也为现实中相关问题的数值模拟与计算提供理论与算法支撑。

The sampling theory seeks to reconstruct or approximate the original signal from sampling values on sampling points. In random sampling, the sampling points are chosen independently from some probability distribution defined on certain bounded domain and the sampling values, depending on the characteristics and performance of sampling devices, may be the signal values on sampling points, the average values of the signal around sampling points, etc. This project studies random sampling and reconstruction in reproducing kernel spaces, which can be divided into three aspects: (1) the stability of random sampling for different available sampling values; (2) the corresponding algorithms that reconstruct or approximate the original signal from finitely many random sampling values, and the performance of the proposed algorithms, such as, stability, convergence rate, the relationship between the sampling size and the probability of achieving corresponding reconstruction error; (3) the statistical inference of the signal, including the hypothesis test of the original signal and the statistical analysis of the reconstructed signal. The knowledge of signal analysis, approximation theory and statistical inference will be used to study the above problems. These studies will enrich the mathematical theory of random sampling, and provide theoretical and algorithmic support for the numerical simulation and calculation of related problems in reality.