Abstract
Dynamical sampling is an emerging paradigm for studying signals that evolve in time. In this chapter we present many of the available results pertaining to dynamical sampling in the finite dimensional setting. We also provide a brief survey of the latest results in the infinite dimensional setting.
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References
R. Aceska, S. Tang, Dynamical sampling in hybrid shift invariant spaces, in  Operator Methods in Wavelets, Tilings, and Frames, ed. by V. Furst, K.A. Kornelson, E.S. Weber. Contemporary Mathematics, vol. 626 (American Mathematical Society, Providence, RI, 2014)
R. Aceska, A. Aldroubi, J. Davis, A. Petrosyan, Dynamical sampling in shift invariant spaces, in  Commutative and Noncommutative Harmonic Analysis and Applications, ed. by A. Mayeli, A. Iosevich, P.E.T. Jorgensen, G. Ólafsson. Contemporary Mathematics, vol. 603 (American Mathematical Society, Providence, RI, 2013), pp. 139–148
E. Acosta-Reyes, A. Aldroubi, I. Krishtal, On stability of sampling-reconstruction models. Adv. Comput. Math.  31, 5–34 (2009)
A. Aldroubi, K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev.  43, 585–620 (2001) (electronic)
A. Aldroubi, I. Krishtal, Robustness of sampling and reconstruction and Beurling-Landau-type theorems for shift-invariant spaces. Appl. Comput. Harmon. Anal.  20, 250–260 (2006)
A. Aldroubi, I. Krishtal, Krylov subspace methods in dynamical sampling (2015). ArXiv:1412.1538
A. Aldroubi, J. Davis, I. Krishtal, Dynamical sampling: time-space trade-off. Appl. Comput. Harmon. Anal.  34, 495–503 (2013)
A. Aldroubi, J. Davis, I. Krishtal, Exact reconstruction of spatially undersampled signals in evolutionary systems. J. Fourier Anal. Appl.  21(1), 11–31 (2015). doi:10.1007/s00041-014-9359-9. ArXiv:1312.3203
A. Aldroubi, C. Cabrelli, U. Molter, S. Tang, Dynamical sampling (2015). ArXiv:1409.8333
R.F. Bass, K. Gröchenig, Relevant sampling of band-limited functions. Illinois J. Math.  57, 43–58 (2013)
J.J. Benedetto, P.J.S.G. Ferreira (eds.),  Modern Sampling Theory. Applied and Numerical Harmonic Analysis (Birkhäuser Boston Inc., Boston, 2001)
T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano, L. Coulot, Sparse sampling of signal innovations. IEEE Signal Process. Mag.  25, 31–40 (2008)
E.J. Candès, C. Fernandez-Granda, Super-resolution from noisy data. J. Fourier Anal. Appl.  19, 1229–1254 (2013)
J. Davis, Dynamical sampling with a forcing term, in  Operator Methods in Wavelets, Tilings, and Frames, ed. by V. Furst, K.A. Kornelson, E.S. Weber. Contemporary Mathematics, vol. 626 (American Mathematical Society, Providence, RI, 2014)
A.G. Garcia, J.M. Kim, K.H. Kwon, G.J. Yoon, Multi-channel sampling on shift-invariant spaces with frame generators. Int. J. Wavelets Multiresolution Inf. Process. Â 10, 1250003, 20 pp (2012)
D. Han, M.Z. Nashed, Q. Sun, Sampling expansions in reproducing kernel Hilbert and Banach spaces. Numer. Funct. Anal. Optim.  30, 971–987 (2009)
J.A. Hogan, J.D. Lakey,  Duration and Bandwidth Limiting. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2012)
P.E.T. Jorgensen, A sampling theory for infinite weighted graphs. Opuscula Math.  31, 209–236 (2011)
M. Liang, J. Du, H. Liu, Spatiotemporal super-resolution reconstruction based on robust optical flow and Zernike moment for video sequences. Math. Probl. Eng. 14 pp. (2013). Art. ID 745752
Y. LyubarskiÄ, W.R. Madych, The recovery of irregularly sampled band limited functions via tempered splines. J. Funct. Anal.  125, 201–222 (1994)
M.Z. Nashed, Q. Sun, Sampling and reconstruction of signals in a reproducing kernel subspace of \(L^{p}(\mathbb{R}^{d})\). J. Funct. Anal.  258, 2422–2452 (2010)
A. Papoulis, Generalized sampling expansion, in  IEEE Transactions on Circuits and Systems, CAS-24 (1977), pp. 652–654
Q. Sun, Local reconstruction for sampling in shift-invariant spaces. Adv. Comput. Math.  32, 335–352 (2010)
P.P. Vaidyanathan, V.C. Liu, Classical sampling theorems in the context of multirate and polyphase digital filter bank structures. IEEE Trans. Acoust. Speech Signal Process.  36, 1480–1495 (1988)
N. Wiener, Â Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications (The Technology Press of the Massachusetts Institute of Technology, Cambridge, 1949)
F. Xue, F. Luisier, T. Blu, Multi-Wiener SURE-LET deconvolution. IEEE Trans. Image Process.  22, 1954–1968 (2013)
Acknowledgements
The research in this chapter is funded by the collaborative NSF ATD grant DMS-1322127 and DMS-1322099. The authors would like to thank the organizers of the FFT at the University of Maryland for the opportunity to present our research to a wide audience of mathematicians and engineers. We are also grateful to S. J. Rose for his continued involvement in our projects.
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Aldroubi, A., Krishtal, I., Weber, E. (2015). Finite Dimensional Dynamical Sampling: An Overview. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 4. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20188-7_9
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