Sampling And Reconstruction For Non-decaying Signals In Mixed Lebesgue Spaces

Sampling theorem is the theoretical basis of modern communication system and one of the most powerful basic tools in signal analysis.It is widely used in many fields such as digital signal processing.In the sampling process,the signal is usually regarded as a function.If the function comes from the mixed Lebesgue space,then we can consider the integrability of each separated variable.For the sampling and reconstruction of non-decaying signals,the growing rate of the signals is controlled by the non-decaying weighting function.This paper mainly discusses and studies the sampling and reconstruction of non-decaying signals according to the characteristics of mixed Lebesgue spaces.In the first chapter,firstly,the research content of this topic is introduced.Secondly,the development process of sampling theory and mixed Lebesgue space theory are summarized.Finally,the existing research results and the research methods and innovations of this paper are summarized.In the second chapter,we first introduce the concept of mixed Lebesgue space and some conclusions that need to be used in the process of proof.Then we get some Riesz-type bounds corresponding to the shift-invariant spaces of non-decaying signals by using the Holder inequality and other related formulas.Finally,we prove that the sampling and reconstruction of signals in L_p,q,1/?(R^d+1)are stable according to the Riesz-type bounds and related properties.In the third chapter,we first introduce the definition of weighted mixed Sobolev space and its corresponding norm,then introduce some properties of Bessel potential kernel,and finally discuss the ideal sampling of non-decaying signals when sampling kernel is not available.In the fourth chapter,we first prove that P?,h is a bounded projector from L₍p,q,-?(R^d+1)onto V_p,q,-?,h(?),and then we prove that the weighted norm of the error between f ?H_p,q,-?^L(R^d+1)and its smooth form J_hf is upper bound.