Skip to main content
SpringerLink
Log in

Generalized Sampling in \({L}^{2}({\mathbb{R}}^{d})\)Shift-Invariant Subspaces with Multiple Stable Generators

  • Chapter
  • First Online:
Multiscale Signal Analysis and Modeling

Abstract

In order to avoid most of the problems associated with classical Shannon’s sampling theory, nowadays, signals are assumed to belong to some shift-invariant subspace. In this work we consider a general shift-invariant space V Φ 2 of L 2(ℝ d) with a set Φ of r stable generators. Besides, in many common situations, the available data of a signal are samples of some filtered versions of the signal itself taken at a sub-lattice of ℤ d. This leads to the problem of generalized sampling in shift-invariant spaces. Assuming that the ℓ 2-norm of the generalized samples of any f∈V Φ 2 is stable with respect to the L 2(ℝ d)-norm of the signal f, we derive frame expansions in the shift-invariant subspace allowing the recovery of the signals in V Φ 2 from the available data. The mathematical technique used here mimics the Fourier duality technique which works for classical Paley–Wiener spaces.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Acosta-Reyes E, Aldroubi A, Krishtal I (2009) On stability of sampling-reconstruction models. Adv Comput Math 31:5–34

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams RA, Fournier JJF (2003) Sobolev spaces. Academic Press, Amsterdam

    MATH  Google Scholar 

  3. Aldroubi A (2002) Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelets spaces. Appl Comput Harmon Anal 13:151–161

    Article  MathSciNet  MATH  Google Scholar 

  4. Aldroubi A, Gröchenig K (2001) Non-uniform sampling and reconstruction in shift-invariant spaces. SIAM Rev 43:585–620

    Article  MathSciNet  MATH  Google Scholar 

  5. Aldroubi A, Sun Q, Tang W-S (2005) Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces. J Fourier Anal Appl 11:215–244

    Article  MathSciNet  MATH  Google Scholar 

  6. Aldroubi A, Unser M (1994) Sampling procedure in function spaces and asymptotic equivalence with Shannon’s sampling theorem. Numer Funct Anal Optim 15:1–21

    Article  MathSciNet  MATH  Google Scholar 

  7. Aldroubi A, Unser M, Eden M (1992) Cardinal spline filters: stability and convergence to the ideal sinc interpolator. Signal Process 28:127–138

    Article  MATH  Google Scholar 

  8. Boor C, DeVore R, Ron A (1994) Approximation from shift-invariant subspaces in L 2(ℝ d). Trans Am Math Soc 341:787–806

    MATH  Google Scholar 

  9. Boor C, DeVore R, Ron A (1994) The structure of finitely generated shift-invariant spaces in L 2(ℝ d). J Funct Anal 119:37–78

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen W, Itoh S, Shiki J (2002) On sampling in shift invariant spaces. IEEE Trans Signal Process 48:2802–2810

    MathSciNet  MATH  Google Scholar 

  11. Christensen O (2003) An introduction to frames and Riesz bases. Birkhäuser, Boston

    MATH  Google Scholar 

  12. Dahmen W, Han B, Jia RQ, Kunoth A (2000) Biorthogonal multiwavelets on the interval: cubic Hermite spline. Constr Approx 16:221–259

    Article  MathSciNet  MATH  Google Scholar 

  13. DeVore R, Lorentz G (1993) Constructive approximation. Springer, Berlin

    Book  MATH  Google Scholar 

  14. García AG, Pérez-Villalón G (2006) Dual frames in L 2(0, 1) connected with generalized sampling in shift-invariant spaces. Appl Comput Harmon Anal 20:422–433

    Article  MathSciNet  MATH  Google Scholar 

  15. García AG, Pérez-Villalón G (2008) Approximation from shift-invariant spaces by generalized sampling formulas. Appl Comput Harmon Anal 24:58–69

    Article  MathSciNet  MATH  Google Scholar 

  16. García AG, Pérez-Villalón G (2009) Multivariate generalized sampling in shift-invariant spaces and its approximation properties. J Math Anal Appl 355:397–413

    Article  MathSciNet  MATH  Google Scholar 

  17. García AG, Hernández-Medina MA, Pérez-Villalón G (2008) Generalized sampling in shift-invariant spaces with multiple stable generators. J Math Anal Appl 337:69–84

    Article  MathSciNet  MATH  Google Scholar 

  18. García AG, Muñoz-Bouzo MJ, Pérez-Villalón G (2011) Regular multivariate sampling and approximation in L p shift-invariant spaces. J Math Anal Appl 380:607–627

    Article  MathSciNet  MATH  Google Scholar 

  19. García AG, Pérez-Villalón G, Portal A (2005) Riesz bases in L 2(0, 1) related to sampling in shift invariant spaces. J Math Anal Appl 308:703–713

    Article  MathSciNet  MATH  Google Scholar 

  20. García AG, Kim JM, Kwon KH, Pérez-Villalón G (2008) Aliasing error of sampling series in wavelets subspaces. Numer Funct Anal Optim 29:126–144

    Article  MathSciNet  MATH  Google Scholar 

  21. García AG, Kim JM, Kwon KH, Yoon GJ (2012) Multi-channel sampling on shift-invariant spaces with frame generators. Int J Wavelets Multiresolut Inform Process 10(1):41–60

    Google Scholar 

  22. Gröchenig K (2001) Foundations of time-frequency analysis. Birkhäuser, Boston

    MATH  Google Scholar 

  23. Higgins JR (1996) Sampling theory in fourier and signal analysis: foundations. Oxford University Press, Oxford

    MATH  Google Scholar 

  24. Hogan JA, Lakey JD (2005) Sampling in principal shif-invariant spaces. In: Time-frequency and time-scale methods. Birkhäuser, Boston

    Google Scholar 

  25. Hong YM, Kim JM, Kwon KH, Lee EM (2007) Channeled sampling in shift-invariant spaces. Int J Wavelets Multiresolut Inform Process 5:753–767

    Article  MathSciNet  MATH  Google Scholar 

  26. Horn RA, Johnson CR (1999) Matrix analysis. Cambridge University Press, Cambridge

    Google Scholar 

  27. Jia RQ, Micchelli CA (1991) Using the refinement equations for the construction of pre-wavelets II: Powers of two. In: Laurent PJ, Le Méhauté A, Schumaker LL (eds) Curves and surfaces. Academic Press, New York, pp 209–246

    Google Scholar 

  28. Jia RQ, Shen Z (1994) Multiresolution and wavelets. Proc Edinbur Math Soc 37:271–300

    Article  MathSciNet  MATH  Google Scholar 

  29. Keinert F (2004) Wavelets and multiwavelets. Chapman & Hall/CRC, Boca Raton FL

    MATH  Google Scholar 

  30. Kang S, Kwon KH (2011) Generalized average sampling in shift-invariant spaces. J Math Anal Appl 377:70–78

    Article  MathSciNet  MATH  Google Scholar 

  31. Kang S, Kim JM, Kwon KH (2010) Asymmetric multi-channel sampling in shift-invariant spaces. J Math Anal Appl 367:20–28

    Article  MathSciNet  MATH  Google Scholar 

  32. Kim KH, Kwon KH (2008) Sampling expansions in shift-invariant spaces. Int J Wavelets Multiresolut Inform Process 6:223–248

    Article  MathSciNet  MATH  Google Scholar 

  33. Lei JJ, Jia RQ, Cheney EW (1997) Approximation from shift-invariant spaces by integral operators. SIAM J Math Anal 28:481–498

    Article  MathSciNet  MATH  Google Scholar 

  34. Nashed MZ, Sun Q, Tang W-S (2009) Average sampling in L 2. Compt Rend Acad Sci Paris, Ser I 347:1007–1010

    Article  MathSciNet  MATH  Google Scholar 

  35. Papoulis A (1977) Generalized sampling expansion. IEEE Trans Circuits Syst 24:652–654

    Article  MathSciNet  MATH  Google Scholar 

  36. Ron A, Shen Z (1995) Frames and stable bases for shift-invariant subspaces of L 2(ℝ d). Can J Math 47:1051–1094

    Article  MathSciNet  MATH  Google Scholar 

  37. Saitoh S (1997) Integral transforms, reproducing kernels and their applications. Longman, Essex

    MATH  Google Scholar 

  38. Selesnick IW (1999) Interpolating multiwavelet bases and the sampling theorem. IEEE Trans Signal Process 47:1615–1620

    Article  MathSciNet  MATH  Google Scholar 

  39. {S}ikić H, Wilson EN (2011) Lattice invariant subspaces and sampling. Appl Comput Harmon Anal 31:26–43

    Google Scholar 

  40. Sun Q (2006) Nonuniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J Math Anal 38:1389–1422

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun Q (2010) Local reconstruction for sampling in shift-invariant spaces. Adv Comput Math 32:335–352

    Article  MathSciNet  MATH  Google Scholar 

  42. Sun W (1999) On the stability of multivariate trigonometric systems. J Math Anal Appl 235:159–167

    Article  MathSciNet  MATH  Google Scholar 

  43. Sun W, Zhou X (1999) Sampling theorem for multiwavelet subspaces Chin Sci Bull 44:1283–1285

    Article  MathSciNet  MATH  Google Scholar 

  44. Unser M (1999) Splines: a perfect fit for signal and image processing. IEEE Signal Process Mag 16:22–38

    Article  Google Scholar 

  45. Unser M (2000) Sampling 50 years after Shannon. Proc IEEE 88:569–587

    Article  Google Scholar 

  46. Vaidyanathan PP (1993) Multirate systems and filter banks. Prentice-Hall, Englewood Hills

    MATH  Google Scholar 

  47. Walter GG (1992) A sampling theorem for wavelet subspaces. IEEE Trans Inform Theor 38:881–884

    Article  MATH  Google Scholar 

  48. Wojtaszczyk P (1997) A mathematical introduction to wavelets. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  49. Xian J, Li S (2007) Sampling set conditions in weighted multiply generated shift-invariant spaces and their applications. Appl Comput Harmon Anal 23:171–180

    Article  MathSciNet  MATH  Google Scholar 

  50. Xian J, Sun W (2010) Local sampling and reconstruction in shift-invariant spaces and their applications in spline subspaces. Numer Funct Anal Optim 31:366–386

    Article  MathSciNet  MATH  Google Scholar 

  51. Xian J, Luo SP, Lin W (2006) Weighted sampling and signal reconstruction in spline subspace. Signal Process 86:331–340

    Article  MATH  Google Scholar 

  52. Zayed AI (1993) Advances in Shannon’s sampling theory. CRC Press, Boca Raton

    MATH  Google Scholar 

  53. Zhou X, Sun W (1999) On the sampling theorem for wavelet subspaces. J Fourier Anal Appl 5:347–354

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very pleased to dedicate this work to Professor Gilbert. G.Walter on the occasion of his 80th birthday. Professor Walter’s research andmentorship have, over the years, inspired and influenced many mathematiciansthroughout the world; we are fortunate to be three of these mathematicians.

This work has been supported by the grant MTM2009–08345 from the Spanish Ministerio de Ciencia e Innovación (MICINN).

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911, Leganés (Madrid), Spain

    H. R. Fernández-Morales & A. G. García

  2. Departamento de Matemática Aplicada, E.U.I.T.T., U.P.M., 28031, Madrid, Spain

    G. Pérez-Villalón

Authors
  1. H. R. Fernández-Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. García
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Pérez-Villalón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. García .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Ohio University, Ohio University 1, Athens, 45701, Ohio, USA

    Xiaoping Shen

  2. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614, Illinois, USA

    Ahmed I. Zayed

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer New York

About this chapter

Cite this chapter

Fernández-Morales, H.R., García, A.G., Pérez-Villalón, G. (2013). Generalized Sampling in \({L}^{2}({\mathbb{R}}^{d})\)Shift-Invariant Subspaces with Multiple Stable Generators. In: Shen, X., Zayed, A. (eds) Multiscale Signal Analysis and Modeling. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4145-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-4145-8_3

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-4144-1

  • Online ISBN: 978-1-4614-4145-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation