Random Sampling Problem In Reproducing Kernel Space

With the rapid development of the information age,the sampling theory of signals is playing an increasingly important role.In particular,random sampling,a rather active sampling method in sampling theory,has played a pivotal role in image processing,compressed sensing and learning theory.Furthermore,signals often exist in both the time domain and the space domain,that is,most of them are time varying signals.Since mixed Lebesgue spaces have separate integrability for different variables,it can be used precisely to model and measure time-varying signals.In this paper,random sampling of signals is studied based on reproducing kernel space,focusing on random sampling and reconstruction of energy concentrated signals in reproducing kernel subspace of mixed Lebesgue space:1.The random sampling stability of signals in the weighted reproducing kernel subspace of classical Lebesgue space is mainly studied.The sample set follows the general probability distribution onR~d and the relatively strong condition of symmetry of kernel function is no longer considered.Firstly,a weighted finite dimensional subspace is used to approximate a weighted reproducing kernel subspace.Secondly,it is proved that the random sampling stability of the signal whose energy is concentrated on the bounded cube is valid with high probability when the sample size is large enough.2.The sampling and reconstruction of energy concentrated signals in reproducing kernel subspace of mixed Lebesgue space is studied.Firstly,the iterative reconstruction algorithm of the reproducing kernel subspace is revisited and reformulated based on mixed Lebesgue space.Secondly,the weighted stability inequality is established and a good algorithm is proposed to approximate the energy concentrated signals in the reproducing kernel space.Finally,based on randomly selected samples,it is proved that the energy concentrated signals in the reproducing kernel space can be approximated with high probability.