## Abstract

In practical applications the sole reconstruction of signals by its samples is sometimes not sufficient. Often, some processed version of the signal is of interest and must be approximated by using only the samples of the signal. In this paper, the possible reconstruction kernels are characterized. Then, the convergence behavior of general approximation processes for translation invariant, linear, and bounded operators is analyzed for signals in the Paley-Wiener space PW^{1} Π and these kernels. It is shown that the Hilbert transform is a universal operator in the sense that the peak value of all possible approximation processes diverges unboundedly for some signal in PW^{1} Π, regardless of the oversampling factor and the kernel. Furthermore, for all approximation processes and all points in time, there exists an operator such that the approximation process diverges in this point. The results are compared to the approximation behavior of the Hilbert transform, operating on continuous-time signals. Moreover, a simple criterion based on the exponential function as test signal is developed for answering the question of whether or not an approximation process is convergent for a given operator.

Original language | English |
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Pages (from-to) | 119-153 |

Number of pages | 35 |

Journal | Sampling Theory in Signal and Image Processing |

Volume | 9 |

Issue number | 1-3 |

State | Published - 2010 |

## Keywords

- Reconstruction process
- Sampling series
- Sampling-based signal processing
- System representation
- Uniform approximation