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

TL;DR

This paper introduces a local sampling and approximation method for operators with bandlimited Kohn-Nirenberg symbols, enabling localized operator recovery using truncated, mollified signals, addressing non-locality issues in existing theories.

Contribution

It develops tools for local operator approximation using truncated and mollified signals, filling a gap in the theory of operator sampling.

Findings

Local approximations are achievable with truncated, mollified delta trains.

Discrete measurements can be localized to yield local operator approximations.

The concept of localization for bandlimited Kohn-Nirenberg operators is clarified.

Abstract

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