Arbitrarily Slow Convergence For Non-uniformly Sampled Reconstructions

In signal processing some continuous-time band-limited signals can be reconstructed by using discrete-time samples.Shannon sampling theorem is one of the most famous results in sampling theory.It provides the exact reconstruction of a bandlimited signal from its samples at the Nyquist rate by a cardinal series.This equidistant sampling reconstruction is not the only possible one,because of the variety of signals.Nonuniform sampling occurs as frequently in practice and is sometimes known to give better results.In recent years,many achievements have been given in the study of nonuniform sampling patterns.However these reconstruction patterns are not applied widely in practice because of their slow convergence.Thus it is of theoretical value to analyze the convergence rate of reconstruction from the nonuniform sampling series.In this work,the convergence rate of nonuniform reconstruction is studied by the theory on the slow convergence of operator sequences and the main conclusions are proved as follows:the nonuniform sampling reconstruction process is arbitrarily slow convergence.Specifically,the main contents of this paper are divided into three parts:(1)The basic pattern of the nonuniform sampling reconstruction is analysed,i.e.,for any bandlimited signal f,it can be reconstructed by its samples at points{tk}k?Z from the following cardinal series f(t)=?=-?+?f(tk)?k(t),t?R,where?k are reconstruction kernel function,determined by sine-type function.And we introduce some necessary results about nonuniform sample and the theory of slow convergence of operator sequences.(2)we analyze the convergence rate of sampling patterns that are made of the zeros of sine-type functions.It is shown that there exists a sequence of "arbitrarily slow" reconstruction operators in the reconstruction process from nonuniform samples.Specifically,for any positive sequence ?(n)?0,there exists a bandlimited signal f such that the n-th truncation error of its cardinal series is larger than ?(n)for all n ?R,where the truncation errors are measured in Lp(1<p<?)norms.Moreover,we analyze the convergence rate of the reconstruction operator sequence with acceleration techniques.It is also proved that the acceleration techniques by over-sampling and convergence factor cannot change the convergence rate,which is still "arbitrarily slow".(3)We construct real signals conforming to the theoretical results and depict plots of signals.In the numerical examples we can construct a signal such that the convergence rate of its reconstruction process is slower than any ?(n)by PlancherelPolya inequality to emphasize why theoretical results are useful.