Random Sampling And Reconstruction Of Non-decaying Signals From Multiply Generated Shift-invariant Spaces

Sampling theory has a rich history,starting from the classical Whitaker-Shannon-Cottel's Nikov sampling theorem.The main fact behind the sampling theorem is that the band-limited signal lives in a displacement invariant space generated by the CINC function.The sampling theory is extended to the general displacement invariant space generated by splines or wavelets,which have better localization than the ideal sinusoidal kernel.Although a great deal of work has been devoted to sampling shifted invariant Spaces,the general theory of sampling unattenuated signals seems to be missing.However,in the absence of a prefilter,the ideal sampling of LP signal weighting cannot be carried out stably.None of the existing theories applies to the ideal sampling of non-decreasing general functions.Intuitively,however,there seems to be no fundamental reason to prevent one from sampling a continuous signal,even if it is not decaying.In recent years,many scholars have done a lot of research on the reconstruction algorithms of various function spaces and the conditions of stable sampling sets.There is a classic in the field of communication sampling theorem to refactor-Shannon sampling theorem,the reconfigurable Shannon sampling theorem,random sampling and reconstruction for the attenuation of the signal with further research mainly expounds a continuous signal f(x)into a discrete signal f(x_j),discrete signal f(x_j)to reconstruct(recovery)to the process of the continuous signal f(x).Random sampling and reconstruction of non-attenuating signals is a challenging problem.In this paper,we study the random sampling and reconstruction of non-decaying signals based on multiple generated translational invariant Spaces,the sampling set inequality and reconstruction formula of the following form are valid in the sense of probability:By estimating the attenuation of the signal generated based on translation-invariant space sampling set inequality probability upper bound,we give the reconstruction formula of probability sense.We consider both noise and spatial weighting,which is not previously available,and our results generalize the previous results.