Skip to main content
Log in

Local Sampling and Approximation of Operators with Bandlimited Kohn–Nirenberg Symbols

  • Published:
Constructive Approximation Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 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.

  2. 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 \(\int \varphi _\delta =1\).

  3. 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 \(\beta _1\ge 1 +2\delta /\Omega \).

References

  1. Bajwa, W.U., Gedalyahu, K., Eldar, Y.C.: Identification of parametric underspread linear systems and super-resolution radar. IEEE Trans. Signal Process. 59(6), 2548–2561 (2011)

    Article  MathSciNet  Google Scholar 

  2. Bello, P.A.: Measurement of random time-variant linear channels. IEEE Trans. Commun. 15, 469–475 (1969)

    MATH  Google Scholar 

  3. Benedetto, J.J., Oktay, O.: Pointwise comparison of PCM and \(\Sigma \Delta \) quantization. Constr. Approx. 32(1), 131–158 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Sigma-delta quantization and finite frames. IEEE Trans. Inf. Theory 52, 1990–2005 (2006)

    Article  MATH  Google Scholar 

  5. Bényi, Á., Gröchenig, K., Okoudjou, K.A., Rogers, L.G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246(2), 366–384 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Daubechies, I., DeVore, R.: Reconstructing a bandlimited function from very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order. Ann. Math. 158, 679–710 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Deift, P., Güntürk, C.S., Krahmer, F.: An optimal family of exponentially accurate one-bit sigma-delta quantization schemes. Commun. Pure Appl. Math. 64(7), 883–919 (2011)

    Article  MATH  Google Scholar 

  8. Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. Rocky Mt. J. Math. 19, 113–126 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feichtinger, H.G., Gröchenig, K.: Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: Chui, C.K. (ed.) Wavelets. Wavelet Analysis and Applications, vol. 2, pp. 359–397. Academic Press, Boston (1992)

  10. Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)

    Google Scholar 

  11. Gröchenig, K.: Foundations of Time–Frequency Analysis. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  12. Gröchenig, K., Heil, C.: Modulation spaces and pseudodifferential operators. Integr. Equ. Oper. Theory 34(4), 439–457 (1999). MR 1702232 (2001a:47051)

    Article  Google Scholar 

  13. Güntürk, C.S.: One-bit sigma-delta quantization with exponential accuracy. Commun. Pure Appl. Math. 56, 1608–1630 (2003)

    Article  MATH  Google Scholar 

  14. Heckel, R., Bölcskei, H.: Identification of sparse linear operators. IEEE Trans. Inform. Theory 59(12), 7985–8000 (2013)

    Google Scholar 

  15. Kailath, T.: Measurements on time-variant communication channels. IEEE Trans. Inform. Theory 8(5), 229–236 (1962)

    Article  MATH  Google Scholar 

  16. Kozek, W., Pfander, G.E.: Identification of operators with bandlimited symbols. SIAM J. Math. Anal. 37(3), 867–888 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Krahmer, F., Pfander, G.E.: Sampling and quantization of approximately bandlimited operators. (2013, in preparation)

  18. Krahmer, F., Pfander, G.E., Rashkov, P.: Uncertainty principles for timefrequency representations on finite Abelian groups. Appl. Comput. Harmon. Anal. 25, 209–225 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lammers, M., Powell, A.M.: Alternative dual frames for digital-to-analog conversion in sigma-delta quantization. Adv. Comput. Math. 32(1), 73–102 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lawrence, J., Pfander, G.E., Walnut, D.: Linear independence of Gabor systems in finite dimensional vector spaces. J. Fourier Anal. Appl. 11(6), 715–726 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Okoudjou, K.A.: A Beurling–Helson type theorem for modulation spaces. J. Funct. Spaces Appl. 7(1), 33–41 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-Time Signal Processing. Prentice-Hall Signal Processing, 2nd edn. Prentice-Hall, Upper Saddle River (1999)

    Google Scholar 

  23. Petersen, D.P., Middleton, D.: Sampling and reconstruction of wave-number-limited functions in \(N\)-dimensional Euclidean spaces. Inf. Control 5, 279–323 (1962). MR 0151331 (27 #1317)

    Article  MathSciNet  Google Scholar 

  24. Pfander, G.E.: Measurement of time-varying multiple-input multiple-output channels. Appl. Comput. Harmon. Anal. 24, 393–401 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pfander, G.E.: Sampling of operators. J. Fourier Anal. Appl. 19(3), 612–650 (2013)

    Article  MathSciNet  Google Scholar 

  26. Pfander, G.E., Walnut, D.: Sampling and reconstruction of operators (preprint)

  27. Pfander, G.E., Walnut, D.F.: Measurement of time-variant linear channels. IEEE Trans. Inform. Theory 52(11), 4808–4820 (2006)

    Article  MathSciNet  Google Scholar 

  28. Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces \(L_2({ R}^d)\). Can. J. Math. 47(5), 1051–1094 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Strohmer, T.: Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comput. Harmon. Anal. 20(2), 237–249 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus-I. J. Funct. Anal. 207(2), 399–429 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus-II. Ann. Glob. Anal. Geom. 26(1), 73–106 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Toft, J.: Continuity and schatten properties for pseudo-differential operators on modulation spaces. Modern Trends in Pseudo-Differential Operators. Springer, Birkhäuser (2007)

    Chapter  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Felix Krahmer.

Additional information

Communicated by Karlheinz Groechenig.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-014-9228-4

Keywords

Mathematics Subject Classification

Navigation