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 nonlocal. For example, we show that in order to recover a bandlimited operator precisely, the identifier cannot decay in time or 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 is localized in the sense that a local selection of the values yields a local approximation of the operator. Central to our analysis is the conceptualization of the meaning of localization for operators with bandlimited Kohn–Nirenberg symbols.
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Notes
In general terms, operator Paley–Wiener spaces are defined by requiring their members to have bandlimited Kohn–Nirenberg symbols that are in a prescribed weighted and mixed \(L^p\) space [25]. For example, to restrict the attention to bandlimited Hilbert-Schmidt operators, we would consider only operators with square integrable symbols. These form a subset of the operators considered in this paper.
For example, we can choose \(r=\chi _{[0,T)}*\varphi _\delta \), where \(\varphi _\delta \) is an approximate identity, that is, a nonnegative function with \(\varphi _\delta \in \mathcal S(\mathbb {R}), \mathrm{supp }\varphi _\delta \subseteq [-\delta /2,\delta /2]\), and \(\int \varphi _\delta =1\).
Then the Gabor systems \(\{r_{k,l}=\mathcal T_{kT}\mathcal M_{\ell /\beta _2 T}\,r\}_{k,\ell \in \mathbb {Z}}, \{\mathcal T_{n\Omega }\mathcal M_{m/\beta _1 \Omega }\, \widehat{\phi }\}_{m,n\in \mathbb {Z}}\), and \(\{ \Phi _{m,-n,l,-k}=\mathcal T_{(m T L/{\beta _1},\ell L \Omega )}\mathcal M_{(n\Omega , /{\beta _2}, kT)}\}_{m,n,k,\ell \in \mathbb {Z}}\) are tight Gabor frames with \(A=\beta _2/T, A=\beta _1/\Omega \), and \(A=\beta _1\beta _2/(T\Omega )=\beta _1\beta _2 L\), respectively, whenever \(\beta _2\ge 1 +2\delta /T\) and \(\beta _1\ge 1 +2\delta /\Omega \).
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Acknowledgments
The authors thank the anonymous referee for the constructive comments, which greatly improved the paper, and Onur Oktay, who participated in initial discussions on the project. Part of this research was carried out during a sabbatical of G.E.P. and a stay of F.K. at the Department of Mathematics and the Research Laboratory for Electronics at the Massachusetts Institute of Technology. Both are grateful for the support and the stimulating research environment. F.K. acknowledges support by the Hausdorff Center for Mathematics, Bonn. G.E.P. acknowledges funding by the German Science Foundation (DFG) under Grant 50292 DFG PF-4, Sampling Operators.
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Communicated by Karlheinz Groechenig.
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Krahmer, F., Pfander, G.E. Local Sampling and Approximation of Operators with Bandlimited Kohn–Nirenberg Symbols. Constr Approx 39, 541–572 (2014). https://doi.org/10.1007/s00365-014-9228-4
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DOI: https://doi.org/10.1007/s00365-014-9228-4
Keywords
- Operator identification
- Pseudodifferential operators
- Kohn–Nirenberg symbol
- Time frequency localization
- Local approximation
- Tight Gabor frames