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Journal of Function Spaces
Volume 2016, Article ID 4573940, 14 pages
http://dx.doi.org/10.1155/2016/4573940
Research Article

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

1Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés, 28911 Madrid, Spain
2Departamento de Matemáticas Fundamentales, U.N.E.D., Senda del Rey 9, 28040 Madrid, Spain

Received 25 July 2016; Accepted 16 October 2016

Academic Editor: Hans G. Feichtinger

Copyright © 2016 Antonio G. García et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo, “On some sampling-related frames in U-invariant spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 761620, 14 pages, 2013. View at Publisher · View at Google Scholar
  2. H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo, “Generalized sampling: from shift-invariant to U-invariant spaces,” Analysis and Applications, vol. 13, no. 3, pp. 303–329, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. A. G. García and M. Muñoz-Bouzo, “Sampling-related frames in finite U-invariant subspaces,” Applied and Computational Harmonic Analysis, vol. 39, no. 1, pp. 173–184, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. T. Michaeli, V. Pohl, and Y. C. Eldar, “U-invariant sampling: extrapolation and causal interpolation from generalized samples,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2085–2100, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. V. Pohl and H. Boche, “U-invariant sampling and reconstruction in atomic spaces with multiple generators,” IEEE Transactions on Signal Processing, vol. 60, no. 7, pp. 3506–3519, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A. Aldroubi, “Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces,” Applied and Computational Harmonic Analysis, vol. 13, no. 2, pp. 151–161, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. Aldroubi and K. Gröchenig, “Nonuniform sampling and reconstruction in shift-invariant spaces,” SIAM Review, vol. 43, no. 4, pp. 585–620, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. H. R. Fernández-Morales, A. G. García, and G. Pérez-Villalón, “Generalized sampling in L2Rd shift-invariant subspaces with multiple stable generators,” in Multiscale Signal Analysis and Modeling, Lecture Notes in Electrical Engineering, pp. 51–80, Springer, New York, NY, USA, 2012. View at Google Scholar
  10. A. G. García and G. Pérez-Villalón, “Dual frames in L2(0,1) connected with generalized sampling in shift-invariant spaces,” Applied and Computational Harmonic Analysis, vol. 20, no. 3, pp. 422–433, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. G. García, M. A. Hernández-Medina, and G. Pérez-Villalón, “Generalized sampling in shift-invariant spaces with multiple stable generators,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 69–84, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. Kang and K. H. Kwon, “Generalized average sampling in shift invariant spaces,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 70–78, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. C. V. M. van der Mee, M. Z. Nashed, and S. Seatzu, “Sampling expansions and interpolation in unitarily translation invariant reproducing kernel Hilbert spaces,” Advances in Computational Mathematics, vol. 19, no. 4, pp. 355–372, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Q. Sun, “Local reconstruction for sampling in shift-invariant spaces,” Advances in Computational Mathematics, vol. 32, no. 3, pp. 335–352, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. W. Sun and X. Zhou, “Average sampling in spline subspaces,” Applied Mathematics Letters, vol. 15, no. 2, pp. 233–237, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. Sun and X. Zhou, “Average sampling in shift invariant subspaces with symmetric averaging functions,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 279–295, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. X. Zhou and W. Sun, “On the sampling theorem for wavelet subspaces,” The Journal of Fourier Analysis and Applications, vol. 5, no. 4, pp. 347–354, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  18. H. R. Fernández-Morales, A. G. García, M. J. Muñoz-Bouzo, and A. Ortega, “Finite sampling in multiple generated U-invariant subspaces,” IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 2203–2212, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. G. García, A. Miguel, and A. Ibort, “Towards a quantum sampling theory: the case of finite groups,” https://arxiv.org/abs/1510.08134v1.
  20. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, Pa, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  21. H. Stark, “Polar, spiral, and generalized sampling and interpolation,” in Advanced topics in Shannon Sampling and Interpolation Theory, R. J. Marks II, Ed., pp. 185–207, Springer, New York, NY, USA, 1993. View at Google Scholar
  22. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Transactions of the American Mathematical Society, vol. 72, pp. 341–366, 1952. View at Publisher · View at Google Scholar · View at MathSciNet
  23. I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” Journal of Mathematical Physics, vol. 27, no. 5, pp. 1271–1283, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. P. G. Casazza and G. Kutyniok, Eds., Finite Frames: Theory and Applications, Birkhäuser, Boston, Mass, USA, 2014.
  25. G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, Fla, USA, 1995. View at MathSciNet
  26. A. Bourouihiya, “The tensor product of frames,” Sampling Theory in Signal and Image Processing, vol. 7, no. 1, pp. 65–76, 2008. View at Google Scholar · View at MathSciNet
  27. A. Khosravi and M. S. Asgari, “Frames and bases in tensor product of Hilbert spaces,” International Mathematical Journal, vol. 4, no. 6, pp. 527–537, 2003. View at Google Scholar · View at MathSciNet
  28. C. S. Kubrusly, “A concise introduction to tensor product,” Far East Journal of Mathematical Sciences, vol. 22, no. 2, pp. 137–174, 2006. View at Google Scholar · View at MathSciNet
  29. A. N. Kolmogorov, “Stationary sequences in Hilbert space,” Moscow University Mathematics Bulletin, vol. 2, pp. 1–40, 1941. View at Google Scholar
  30. R. Penrose, “A generalized inverse for matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, no. 3, pp. 406–413, 1955. View at Publisher · View at Google Scholar
  31. A. G. García, “Sampling theory and reproducing kernel Hilbert spaces,” in Operator Theory, D. Alpay, Ed., pp. 87–110, Springer, Basel, Switzerland, 2015. View at Google Scholar