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Multiscale Signal Analysis and Modeling pp 81–104Cite as

Function Spaces for Sampling Expansions

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Abstract

In this chapter, we consider a variety of Hilbert and Banach spaces that admit sampling expansions \(f(t) = \sum\nolimits_{n=1}^{\infty }f({t}_{n}){S}_{n}(t)\), where {S n } n=1 ∞ is a family of functions that depend on the sampling points {t n } n=1 ∞ but not on the function f. Those function spaces, that arise in connection with sampling expansions, include reproducing kernel spaces, Sobolev spaces, Wiener amalgam space, shift-invariant spaces, translation-invariant spaces, and spaces modeling signals with finite rate of innovation. Representative sampling theorems are presented for signals in each of these spaces. The chapter also includes recent results on nonlinear sampling of signals with finite rate of innovation, convolution sampling on Banach spaces, and certain foundational issues in sampling expansions.

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References

  1. Aldroubi A, Gröchenig K (2001) Nonuniform sampling and reconstruction in shift-invariant space, SIAM Rev 43:585–620

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldroubi A, Sun Q, Tang W-S (2001) p-frames and shift-invariant subspaces of L p. J Fourier Anal Appl 7:1–21

    Article  MathSciNet  MATH  Google Scholar 

  3. Aldroubi A, Sun Q, Tang W-S (2004) Nonuniform average sampling and reconstruction in multiply generated shift-invariant spaces. Constr Approx 20:173–189

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 22:215–244

    Article  MathSciNet  Google Scholar 

  5. Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68:337–404

    Article  MathSciNet  MATH  Google Scholar 

  6. Benedetto JJ, Ferreira PJSG (eds.) (2001) Modern sampling theory: mathematics and applications. Birkhuse, Boston

    MATH  Google Scholar 

  7. Benedetto JJ, Zayed AI (eds.) (2003) Sampling, wavelets, and tomography. Birkhauser, Boston

    Google Scholar 

  8. Bi N, Nashed MZ, Sun Q (2009) Reconstructing signals with finite rate of innovation from noisy samples. Acta Appl Math 107:339–372.

    Article  MathSciNet  MATH  Google Scholar 

  9. Butzer PL (1983) A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J Math Res Exposition 3:185–212

    MathSciNet  Google Scholar 

  10. Butzer PL, Splettstöβer W, Stens RL (1988) The sampling theorem and linear prediction in signal analysis. Jahresber Deutsch Math-Verein 90:1–60

    MATH  Google Scholar 

  11. Butzer PL, Stens RL (1992) Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Rev 34:40–53

    Article  MathSciNet  MATH  Google Scholar 

  12. Fornasier M, Rauhut H (2005) Continuous frames, function spaces, and the discretization problem. J Fourier Anal Appl 11:245–287

    Article  MathSciNet  MATH  Google Scholar 

  13. Haddad RA, Parsons TW (1991) Digital signal processing: theory, applications and hardware. Computer Science Press

    Google Scholar 

  14. Han D, Nashed MZ, Sun Q (2009) Sampling expansions in reproducing kernel Hilbert and Banach spaces. Numer Funct Anal Optim 30:971–987

    Article  MathSciNet  MATH  Google Scholar 

  15. Higgins JR (1985) Five short stories about the cardinal series. Bull Am Math Soc 12:45–89

    Article  MathSciNet  MATH  Google Scholar 

  16. Higgins JR (1996) Sampling theory in fourier and signal analysis volume 1: foundations. Oxford University Press, Oxford

    Google Scholar 

  17. Higgins JR, Stens RL (2000) Sampling theory in fourier and signal analysis: volume 2: advanced topics. Oxford Science Publications, Oxford

    Google Scholar 

  18. Ismail M, Nashed Z, Zayed A, Ghaleb A (eds.) (1995) Mathematical analysis, wavelets and signal processing (contemporary mathematics), vol 190. American Mathematical Society, Providence, RI

    Google Scholar 

  19. Jerri JA (1977) The Shannon sampling theorem—its various extensions and applications: a tutorial review. Proc IEEE 65:1565–1596

    Article  MATH  Google Scholar 

  20. Larson D, Massopust P, Nashed Z, Nguyen MC, Papadakis M, Zayed A (eds.) (2008) Frames and operator theory in analysis and signal processing (Contemporary Mathematics), vol 451. American Mathematical Society, Providence, RI

    Google Scholar 

  21. Marks II RJ (1991) Introduction to shannon sampling and interpolation theory. Springer, Berlin

    Book  MATH  Google Scholar 

  22. Marks RJ (1993) Advanced topics in shannon sampling and interpolation theory. Springer, Berlin, Heidelberg

    Book  Google Scholar 

  23. Marvasti FA (ed) (2001) Nonuniform Sampling: theory and practice (information technology: transmission, processing, and storage). Plenum Pub Corp, New York

    Google Scholar 

  24. Nashed MZ, Scherzer O (eds.) (2002) Inverse problems, image analysis and medical imaging (Contemporary Mathematics), vol 313. American Mathematical Society, Providence, RI

    Google Scholar 

  25. Nashed MZ, Sun Q (2010) Sampling and reconstruction of signals in a reproducing kernel subspace of L p(R d). J Funct Anal 258:2422–2452

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Nashed MZ, Sun Q, Xian J Convolution sampling and reconstruction of signals in a reproducing kernel subspace. Proc Am Math Soc (to appear)

    Google Scholar 

  28. Nashed MZ, Walter GG (1991) General sampling theorems for functions in reproducing kernel Hilbert spaces. Math Contr Signals Syst 4:363–390

    Article  MathSciNet  MATH  Google Scholar 

  29. Nashed MZ, Walter GG (1995) Reproducing kernel Hilbert spaces from sampling expansions. Contemporary Math 190:221–226

    Article  MathSciNet  Google Scholar 

  30. Oppenheim AV, Schafer RW (1989) Digital signal processing. Prentice-Hall Inc., New Jersey

    Google Scholar 

  31. Olevskii A, Ulanovskii A (2004) Almost integer translates. Do nice generators exist? J Fourier Anal Appl 10:93–104

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun Q (2006) Non-uniform sampling and reconstruction for signals with finite rate of innovations. SIAM J Math Anal 38:1389–1422

    Article  MathSciNet  MATH  Google Scholar 

  34. Sun Q (2008) Frames in spaces with finite rate of innovation. Adv Comput Math 28:301–329

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun Q (2011) Localized nonlinear functional equations and two sampling problems in signal processing. submitted

    Google Scholar 

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

    Article  Google Scholar 

  38. van der Mee C, Nashed MZ, Seatzu S (2003) Sampling expansions and interpolation in unitarily translation invariant reproducing kernel Hilbert spaces. Adv Comput Math 19:355–372

    Article  MathSciNet  MATH  Google Scholar 

  39. Vetterli M, Marziliano P, Blu T (2002) Sampling signals with finite rate of innovation. IEEE Trans Signal Process 50:1417–1428

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  41. Walter GG, Shen X (2000) Wavelets and other orthogonal systems, 2nd edn. CRC Press, USA

    Google Scholar 

  42. Yao K (1967) Applications of reproducing kernel Hilbert spaces-bandlimited signal models. Inform Contr 11:429–444

    Article  MATH  Google Scholar 

  43. Zayed A (1993) Advances in Shannon’s sampling theory. CRC Press, USA

    MATH  Google Scholar 

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Acknowledgments

The second author is partially supported by the National Science Foundation (DMS-1109063).

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    M. Zuhair Nashed & Qiyu Sun

Authors
  1. M. Zuhair Nashed
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  2. Qiyu Sun
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Correspondence to M. Zuhair Nashed .

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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

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Nashed, M.Z., Sun, Q. (2013). Function Spaces for Sampling Expansions. 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_4

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  • DOI: https://doi.org/10.1007/978-1-4614-4145-8_4

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