In this paper the approximation behavior of the Shannon sampling series is analyzed for the Paley-Wiener space PWπ1, if the samples are disturbed by the non-linear threshold operator. This operator sets all samples, whose absolute value is smaller than some threshold, to zero. It is shown that the peak approximation error can grow arbitrarily large, independently of how small the threshold is. However, if oversampling is applied and an appropriate kernel is chosen, then the re-constructed signal converges to the original signal, uniformly on the whole real axis, as the threshold goes to zero. Furthermore, we analyze the approximation behavior if not the signal itself is to be reconstructed but the output of some stable linear time invariant system. In particular, we show for the Hilbert transform that the peak approximation error is unbounded, even if over-sampling is applied.
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机译：本文分析了Shannon采样系列的近似行为，为Palyy-Wiener Space PW π 1 ，如果样品被非线性阈值操作员扰乱。此操作员设置所有样本，其绝对值小于某个阈值，为零。结果表明，峰值近似误差可以随意增大，独立于阈值的小程度。但是，如果应用过采样并且选择了合适的内核，则重建信号将均匀地在整个实轴上均匀地收敛到原始信号，因为阈值进入零。此外，如果要重建信号本身，则分析近似行为，但是对于一些稳定的线性时间不变系统的输出。特别是，我们向Hilbert变换显示了峰值近似误差是未绑定的，即使应用过度采样也是如此。
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