Approximating signals by fast impulse sampling

Abstract. An impulse sampler multiplies an input signal u by a periodic delta impulse train of period z. If z 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 > 89 then the output is in H -s. Our main result is that as the sampling period z 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" and indicate applications to control theory for distributed parameter systems.