In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results.

翻译：本文讨论了香农采样定理的几种数值实现方法。首先，通过给出香农采样算子范数的精确上下界，展示了经典香农采样和收敛性较差。此外，已知当带限函数的采样值存在噪声时，香农采样级数的收敛性甚至可能完全失效。为克服这些缺陷，可采用过采样并结合合适的窗函数进行正则化处理。此类窗函数既可以在频域中选择，也可以在时域中选择。我们特别强调了这两种方法在误差衰减速率方面的比较。结果表明，采用双曲正弦型窗函数或连续凯泽-贝塞尔窗函数在时域中进行过采样与正则化处理可获得最佳数值结果，由此得到具有局部采样特性的插值逼近。若干数值实验验证了理论结果。