## Abstract

An impulse sampler multiplies an input signal u by a periodic delta impulse train of period Τ. If Τ is small, then the output signal of the sampler (or filtered versions thereof) can be used as an approximation of u. If u belongs to the Sobolev space H^{s} with s>1/2, then the output is in H^{-s}. Our main result is that as the sampling period Τ becomes small, the impulse sampler approximates the identity in the operator norm from H^{s} to H^{-s} (we also give the rate of convergence). We obtain related approximation results in the L^{2} norm, which refer to the situation when filters are connected before and after the impulse sampler (as usually happens in engineering applications). We generalize our results to distributions on ℝ^{n} and indicate applications to control theory for distributed parameter systems.

Original language | English |
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Pages (from-to) | 166-179 |

Number of pages | 14 |

Journal | Mathematics of Control, Signals, and Systems |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1993 |

Externally published | Yes |

## Keywords

- Filter
- Impulse sampler
- Sampling theorem
- Sobolev space