Random Sampling Of Shift-invariant Signals In Mixed Norm Spaces

Random sampling,playing a significant role in learning theory and image processing,is a very important content in sampling theory.The mixed norm Lebesgue spaces have the separate integrability for different variables,which can model and measure time-varying signals widely existing in real life effectively.This paper mainly studies the random sampling problems of shift-invariant signals in mixed norm spaces,emphatically studying the sampling stability of the signals.Note that the physical characteristics of the sampling device,sample value obtained from sampling is often not the exact value of signal at the sample set,but the local average of the signal at the sampling position.Convolution sampling,as a special average sampling,can better model and measure signals.Thus,the first part of this paper mainly studies the convolution random sampling of multiply generated shift-invariant signals in classical Lebesgue spaces,where the sampling set is randomly chosen with a general probability distribution over a cube and the samples are the forms of convolution of the signals.Under some proper conditions for the generators,convolution function and probability density function,we first construct the approximation relationship between finite dimensional subspaces and shift-invariant subspaces in the sense of norm.Then,we establish the sampling stability of energy concentrated signal.Finally,based on the corresponding convolution random samples,a reconstruction algorithm for the signals in finite dimensional subspaces is established.In the second part of this paper,the random sampling theory of signals in single generated shift-invariant subspaces of mixed norm Lebesgue spaces is studied.Under the conditions of compact support and shift stability of the generator,we construct the sampling stability of the energy concentration signal and give the probability estimation by means of the covering number,Bernstein inequality and the related properties of random variables.In the third part of this paper,the random sampling theory of signals in multiply generated shift-invariant subspaces of mixed norm Lebesgue spaces is studied.Under the condition that the generators satisfy decay property and shift stability,we first establish the sampling stability of energy concentrated signals in shift-invariant subspaces.At last,a reconstruction algorithm based on random samples is given for signals in a finite dimensional subspace and the probability estimation is given.