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Recent sampling theorems allow for the recovery of operators with bandlimitedKohn-Nirenberg symbols from their response to a single discretely supportedidentifier signal. The available results are inherently non-local. For example,we show that in order to recover a bandlimited operator precisely, theidentifier cannot decay in time nor in frequency. Moreover, a concept of localand discrete representation is missing from the theory. In this paper, wedevelop tools that address these shortcomings.  We show that to obtain a local approximation of an operator, it is sufficientto test the operator on a truncated and mollified delta train, that is, on acompactly supported Schwarz class function. To compute the operatornumerically, discrete measurements can be obtained from the response functionwhich are localized in the sense that a local selection of the values yields alocal approximation of the operator.  Central to our analysis is to conceptualize the meaning of localization foroperators with bandlimited Kohn-Nirenberg symbol.

Local sampling and approximation of operators with bandlimited Kohn-Nirenberg symbols