关键词: Shannon重构, 慢收敛, 过采样

Abstract: Shannon sampling theorem is an important conclusion in signal processing. It states that the exact reconstruction of a bandlimited signal can be obtained from its samples at Nyquist rate by the cardinal series. It is of theoretical interest to study the convergence rate of the reconstruction of Shannon sampling. In this work, the convergence rate of Shannon's reconstruction is analysed by using the theory on the slow convergence of operator sequences. It is provided that Shannon's reconstruction consists of a sequence of ``arbitrarily slow" convergent operators. Specifically, for any positive sequence $\alpha(n)\to 0$, there exists a bandlimited signal $f$ such that the $n$-th truncation error of its cardinal series is larger than $\alpha(n)$ for all $n$, where the truncation errors are measured in $L^p$ norms, for $1<p<\infty$. It is also shown that the acceleration techniques by over-sampling and convergence factor do not resolve the slowness, which is still ``arbitrarily slowly".

Key words: Shannon reconstruction, slow convergence, over-sampling