February Fourier Talks 2009

David Walnut


Operator Identification and Sampling


Time--invariant communication channels are usually modelled as convolution with a fixed impulse--response function. As the name suggests, such a channel is completely determined by its action on a unit impulse. Time--varying communication channels are modelled as pseudodifferential operators or superpositions of time and frequency shifts. The function or distribution weighting those time and frequency shifts is referred to as the spreading function of the operator.

In this talk we consider the question of whether such operators are identifiable, that is, whether they are completely determined by their action on a single function or distribution. It turns out that the answer is dependent on the size of the support of the spreading function, and that when the operators are identifiable, the input can be chosen as a distribution supported on an appropriately chosen grid. These results provide a sampling theory for operators that can be thought of as a generalization of the classical sampling formula for bandlimited functions.

This is joint work with G. Pfander of Jacobs University, Bremen.