Frame Theory And Sampling Problem In Mixed Norm Space

With the development of information technology,digital signal processing plays a more and more important role.The extraction and recovery of signal has been paid much attention by researchers,so the study of signal sampling and reconstruction has become a hot topic in the current research.Research on signals is mainly carried out in the in the classical Lebesgue space,that is,the signals in the time domain or the space domain.However,signals in real life mostly exist in the form of time domain and space domain,that is,time space signals,also known as time varying signals.Mixed Lebesgue spaces can be used to simulate and measure time varying signals.Therefore,it is of practical significance to study the sampling theory in mixed Lebesgue space.In this paper,the frame theory and sampling problems in shift invariant subspaces of mixed Lebesgue spaces are studied.The main research contents are as follows:1.The frame theory of shift invariant subspace of mixed Lebesgue space is studied.First,the equivalence conditions of the(p,q)-frame are given in the shift invariant subspaces of mixed Lebesgue spaces.Secondly,it is proved that the shift invariant subspace is closed under the frame equivalence condition.2.We mainly study frame-based nonuniform average sampling and reconstruction in the multi-generated shift invariant subspaces of mixed Lebesgue spaces.First,some basic lemmas are established based on the frame.Secondly,an iterative approximation projection algorithm with exponential convergence is established for two average sampling functionals.Finally,estimated rate of convergence.