Average Sampling And Reconstruction Of Signals In Reproducing Kernel Spaces

Sampling problems are concerning of how to utilize the values of the sampling points to restore or approximate the original signal. Sampling and reconstruction which have a major impact on the development of the modern electronic industry technology are the bases of the signal processing. This thesis mainly studies the average sampling and the reconstruction of the deterministic signals and the random signals in the reproducing kernel space. The main contents are as follows:The chapter one discusses the average sampling and the reconstruction of the deterministic signals in the weighted reproducing kernel space. Firstly, the sampling stability for two kinds of the average sampling functionals are established. Secondly, a general iterative reconstruction algorithm is given based on the probability measure,which provides a unified treatment for the iterative approximation projection reconstruction algorithm and iterative frame reconstruction algorithm. Finally, the average error and the asymptotic pointwise error estimates are presented for reconstructing a signal from its samples corrupted by the white noise.The chapter two studies the average sampling and quasi-optimal approximation of the deterministic signals in the reproducing kernel spaces of homogeneous type. Firstly, the pre-reconstruction operator based on finite average samples and the probability measure is proposed and its bounded property is checked. Then, the corresponding sampling stability and an iterative reconstruction algorithm with exponential convergence are established for recovering signals in a subspace of homogeneous reproducing kernel space. Finally, we prove that the proposed algorithm also provides a quasi-optimal approximation to signals in a reproducing kernel space of homogeneous type.The third chapter studies the average sampling and the reconstruction of the stochastic signals in the weighted reproducing kernel space. Firstly, a uniform convergence result for recovering the deterministic weighted reproducing kernel signals by an iterative reconstruction algorithm is established. Then, we prove that the quadratic sum of the corresponding reconstructed functions is uniformly bounded. Finally, the mean square convergence for recovering a weighted reproducing kernel stochastic signal is given under some decay conditions for autocorrelation function, which can be removed for Hilbert space.