# On the Behavior of the Threshold Operator for Bandlimited Functions

## Abstract

One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space \(\mathcal {PW}_{\pi}^{1}\) is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in \(\mathcal {PW}_{\pi}^{1}\), lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the *L* ^{∞}-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling.

## Keywords

Shannon sampling series Linear time invariant system Threshold operator Paley–Wiener space## Mathematics Subject Classification

94A20 94A12## Notes

### Acknowledgements

The authors would like to thank Przemysław Wojtaszczyk for discussions on greedy approximations at the Stobl’11 conference and Ingrid Daubechies for discussions on quantization and oversampling at the Stobl’11 conference and the “Applied Harmonic Analysis and Sparse Approximation” workshop at the Mathematisches Forschungsinstitut Oberwolfach in 2012. We would also like to thank the reviewer, who kindly provided references [14, 15, 16, 17].

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