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若干数值实现方法。首先通过给出香农采样算子范数的严格上下界，展示了经典香农采样和的收敛性较差。此外，已知当带限函数采样值存在噪声时，香农采样级数的收敛甚至可能完全失效。为克服这些缺陷，可采用过采样并结合适当的窗函数进行正则化处理。此类窗函数既可在频域也可在时域内选取。我们特别着重比较了这两种方法在误差衰减速率方面的表现。结果表明，采用$\sinh$型窗函数或连续Kaiser-Bessel窗函数在时域进行过采样与正则化可获得最佳数值结果，这类方法能生成具有局部采样特性的插值逼近。多项数值实验验证了理论分析结果。