Aldroubi, A.; Cabrelli, C.; Molter, U.; Tang, S."Dynamical sampling" (2017) Applied and Computational Harmonic Analysis. 42(3):378-401
Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


Let Y={f(i),Af(i),…,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(I) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states Alf. We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, although Y can be complete, using the Müntz–Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H2(D). Finally, using the recently proved Kadison–Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite. © 2015 Elsevier Inc.


Documento: Artículo
Título:Dynamical sampling
Autor:Aldroubi, A.; Cabrelli, C.; Molter, U.; Tang, S.
Filiación:Department of Mathematics, Vanderbilt University, Nashville, TN 37240-0001, United States
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, Buenos Aires, 1428, Argentina
CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
Palabras clave:Carleson's theorem; Feichtinger conjecture; Frames; Müntz–Szász Theorem; Reconstruction; Sampling theory; Sub-sampling; Harmonic analysis; Image reconstruction; Carleson's theorem; Feichtinger conjecture; Frames; Sampling theory; Sub-sampling; Problem solving
Página de inicio:378
Página de fin:401
Título revista:Applied and Computational Harmonic Analysis
Título revista abreviado:Appl Comput Harmonic Anal


  • Adcock, B., Hansen, A.C., A generalized sampling theorem for stable reconstructions in arbitrary bases (2012) J. Fourier Anal. Appl., 18 (4), pp. 685-716. , MR 2984365
  • Aldroubi, A., Baskakov, A., Krishtal, I., Slanted matrices, Banach frames, and sampling (2008) J. Funct. Anal., 255 (7), pp. 1667-1691. , MR 2442078 (2010a:46059)
  • Aldroubi, A., Davis, J., Krishtal, I., Dynamical sampling: time-space trade-off (2013) Appl. Comput. Harmon. Anal., 34 (3), pp. 495-503. , MR 3027915
  • Aldroubi, A., Davis, J., Krishtal, I., Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off (2015) J. Fourier Anal. Appl., 21, pp. 11-31
  • Aldroubi, A., Gröchenig, K., Nonuniform sampling and reconstruction in shift-invariant spaces (2001) SIAM Rev., 43 (4), pp. 585-620. , (electronic), MR 1882684 (2003e:94040)
  • Antezana, J., (2014), Private communication; Bass, R.F., Gröchenig, K., Relevant sampling of bandlimited functions (2013) Illinois J. Math., 57 (1), pp. 43-58
  • Benedetto, J.J., Ferreira, P.J.S.G., (2001) Modern Sampling Theory, Applied and Numerical Harmonic Analysis, , Birkhäuser Boston Inc. Boston, MA MR 1865678 (2003a:94003)
  • Bratteli, O., Jorgensen, P., Wavelets Through a Looking Glass, The World of the Spectrum (2002) Applied and Numerical Harmonic Analysis, , Birkhäuser Boston Inc. Boston, MA MR 1913212 (2003i:42001)
  • Cahill, J., Casazza, P.G., Li, S., Non-orthogonal fusion frames and the sparsity of fusion frame operators (2012) J. Fourier Anal. Appl., 18 (2), pp. 287-308. , MR 2898730
  • Candès, E.J., Romberg, J.K., Tao, T., Stable signal recovery from incomplete and inaccurate measurements (2006) Comm. Pure Appl. Math., 59 (8), pp. 1207-1223. , MR 2230846 (2007f:94007)
  • Casazza, P.G., Christensen, O., Lindner, A.M., Vershynin, R., Frames and the Feichtinger conjecture (2005) Proc. Amer. Math. Soc., 133 (4), pp. 1025-1033. , (electronic), MR 2117203 (2006a:46024)
  • Casazza, P.G., Kutyniok, G., Li, S., Fusion frames and distributed processing (2008) Appl. Comput. Harmon. Anal., 25 (1), pp. 114-132. , MR 2419707 (2009d:42094)
  • Casazza, P.G., Crandell Tremain, J., The Kadison–Singer problem in mathematics and engineering (2006) Proc. Natl. Acad. Sci. USA, 103 (7), pp. 2032-2039. , (electronic), MR 2204073 (2006j:46074)
  • Conway, J.B., A Course in Functional Analysis (1985) Graduate Texts in Mathematics, 96. , Springer-Verlag New York MR 768926 (86h:46001)
  • Currey, B., Mayeli, A., Gabor fields and wavelet sets for the Heisenberg group (2011) Monatsh. Math., 162 (2), pp. 119-142. , MR 2769882 (2012d:42069)
  • Daubechies, I., Ten Lectures on Wavelets (1992) CBMS-NSF Regional Conference Series in Applied Mathematics, 61. , Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA MR 1162107 (93e:42045)
  • Farrell, B., Strohmer, T., Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group (2010) J. Operator Theory, 64 (1), pp. 189-205. , MR 2669435
  • Garcia, A.G., Kim, J.M., Kwon, K.H., Yoon, G.J., Multi-channel sampling on shift-invariant spaces with frame generators (2012) Int. J. Wavelets Multiresolut. Inf. Process., 10 (1). , 20 pp., MR 2905208
  • Gröchenig, K., Localization of frames, Banach frames, and the invertibility of the frame operator (2004) J. Fourier Anal. Appl., 10 (2), pp. 105-132. , MR 2054304 (2005f:42086)
  • Gröchenig, K., Leinert, M., Wiener's lemma for twisted convolution and Gabor frames (2004) J. Amer. Math. Soc., 17 (1), pp. 1-18. , (electronic), MR 2015328 (2004m:42037)
  • Han, D., Larson, D., Frame duality properties for projective unitary representations (2008) Bull. Lond. Math. Soc., 40 (4), pp. 685-695. , MR 2441141 (2009g:42057)
  • Deguang Han, Zuhair Nashed, M., Sun, Q., Sampling expansions in reproducing kernel Hilbert and Banach spaces (2009) Numer. Funct. Anal. Optim., 30 (9-10), pp. 971-987. , MR 2589760 (2010m:42062)
  • Heil, C., A basis theory primer (2011) Applied and Numerical Harmonic Analysis, , expanded ed. Birkhäuser/Springer New York MR 2744776 (2012b:46022)
  • Hernández, E., Weiss, G., A First Course on Wavelets (1996) Studies in Advanced Mathematics, , CRC Press Boca Raton, FL with a foreword by Yves Meyer, MR 1408902 (97i:42015)
  • Hoffman, K., Banach Spaces of Analytic Functions (1988), Dover Publications, Inc. New York Reprint of the 1962 original, MR 1102893 (92d:46066); Hoffman, K., Kunze, R., Linear Algebra (1971), second edition Prentice-Hall, Inc. Englewood Cliffs, N.J. MR 0276251 (43 #1998); Hogan, J.A., Lakey, J.D., Duration and Bandwidth Limiting (2012) Applied and Numerical Harmonic Analysis, , Birkhäuser/Springer New York MR 2883827 (2012m:42001)
  • Hormati, A., Roy, O., Lu, Y.M., Vetterli, M., Distributed sampling of signals linked by sparse filtering: theory and applications (2010) IEEE Trans. Signal Process., 58 (3), pp. 1095-1109
  • Jorgensen, P.E.T., A sampling theory for infinite weighted graphs (2011) Opuscula Math., 31 (2), pp. 209-236. , MR 2747308 (2012d:05271)
  • Lu, Y.M., Dragotti, P.-L., Vetterli, M., Localization of diffusive sources using spatiotemporal measurements (2011) 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton, Sept 2011, pp. 1072-1076
  • Lu, Y.M., Vetterli, M., Spatial super-resolution of a diffusion field by temporal oversampling in sensor networks (2009) IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2009, April 2009, pp. 2249-2252
  • Lyubarskiĭand, Y., Madych, W.R., The recovery of irregularly sampled band limited functions via tempered splines (1994) J. Funct. Anal., 125 (1), pp. 201-222. , MR 1297019 (96d:41013)
  • Mallat, S., A Wavelet Tour of Signal Processing (1998), Academic Press Inc. San Diego, CA MR 1614527 (99m:94012); Marcus, A., Spielman, D., Srivastave, N., Interlacing families ii: mixed characteristic polynomials and the Kadison–Singer problem (2015) Ann. of Math., 182 (1), pp. 327-350
  • Zuhair Nashed, M., Inverse Problems, Moment Problems, Signal Processing: Un Menage a Trois (2011) Mathematics in Science and Technology, pp. 2-19. , World Sci. Publ. Hackensack, NJ MR 2883419
  • Zuhair Nashed, M., Sun, Q., Sampling and reconstruction of signals in a reproducing kernel subspace of Lp(Rd) (2010) J. Funct. Anal., 258 (7), pp. 2422-2452. , MR 2584749 (2011a:60160)
  • Ólafsson, G., Speegle, D., Wavelets, wavelet sets, and linear actions on Rn (2004) Wavelets, Frames and Operator Theory, Contemp. Math., 345, pp. 253-281. , Amer. Math. Soc. Providence, RI MR 2066833 (2005h:42075)
  • Ranieri, J., Chebira, A., Lu, Y.M., Vetterli, M., Sampling and reconstructing diffusion fields with localized sources (2011) 2011 IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP, May 2011, pp. 4016-4019
  • Reise, G., Matz, G., Reconstruction of time-varying fields in wireless sensor networks using shift-invariant spaces: iterative algorithms and impact of sensor localization errors (2010) IEEE Eleventh International Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2010, pp. 1-5
  • Reise, G., Matz, G., Grochenig, K., Distributed field reconstruction in wireless sensor networks based on hybrid shift-invariant spaces (2012) IEEE Trans. Signal Process., 60 (10), pp. 5426-5439
  • Strang, G., Nguyen, T., Wavelets and Filter Banks (1996), Wellesley-Cambridge Press Wellesley, MA MR 1411910 (98b:94003); Strohmer, T., Finite- and infinite-dimensional models for oversampled filter banks (2001) Modern Sampling Theory, Appl. Numer. Harmon. Anal., pp. 293-315. , Birkhäuser Boston Boston, MA MR 1865692
  • Sun, Q., Nonuniform average sampling and reconstruction of signals with finite rate of innovation (2006) SIAM J. Math. Anal., 38 (5), pp. 1389-1422. , (electronic), MR 2286012 (2008b:94049)
  • Sun, Q., Frames in spaces with finite rate of innovation (2008) Adv. Comput. Math., 28 (4), pp. 301-329. , MR 2390281 (2009c:42093)


---------- APA ----------
Aldroubi, A., Cabrelli, C., Molter, U. & Tang, S. (2017) . Dynamical sampling. Applied and Computational Harmonic Analysis, 42(3), 378-401.
---------- CHICAGO ----------
Aldroubi, A., Cabrelli, C., Molter, U., Tang, S. "Dynamical sampling" . Applied and Computational Harmonic Analysis 42, no. 3 (2017) : 378-401.
---------- MLA ----------
Aldroubi, A., Cabrelli, C., Molter, U., Tang, S. "Dynamical sampling" . Applied and Computational Harmonic Analysis, vol. 42, no. 3, 2017, pp. 378-401.
---------- VANCOUVER ----------
Aldroubi, A., Cabrelli, C., Molter, U., Tang, S. Dynamical sampling. Appl Comput Harmonic Anal. 2017;42(3):378-401.