Odeljenje za matematiku, 21. april 2011.

Naredni sastanak Odeljenja za matematiku održaće se u četvrtak, 21. aprila 2011. u sali 2 SANU sa početkom u 14h.

Predavač: Prof. Goetz Pfander, PhD, Jacobs University Bremen

Naslov predavanja: SAMPLING OF OPERATORS

Abstract: The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist, and Kotelnikov, states that a bandlimited function can be recovered from its samples, as long as we use a sufficiently dense sampling grid. Here, we review our recent development of an operator sampling theory which allows for a widening of the classical sampling theorem. In this realm, bandlimited functions are replaced by bandlimited operators, that is, by pseudodifferential operators which have bandlimited Kohn-Nirenberg symbols. Similar to the Nyquist sampling density condition alluded to above, we discuss sufficient and necessary conditions on the bandlimitation of pseudodifferential operators to ensure that they can be recovered by their action on a single distribution. In fact, we show that an operator with Kohn-Nirenberg symbol bandlimited to a Jordan domain of measure less than one can be recovered through its action on a distribution defined on a appropriately chosen sampling grid. Further, an operator with bandlimitation to a Jordan domain of measure larger than one cannot be recovered through its action on any tempered distribution whatsoever, pointing towards a fundamental difference to the classical sampling theorem where a large bandwidth could always be compensated through a sufficiently fine sampling grid. The dichotomy depending on the size of the bandlimitation is related to Heisenberg's uncertainty principle.



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