Skip to main content
SpringerLink
Log in

Multivariate irregular sampling theorem

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result, we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shannon C E. A mathematical theory of communication. Bell System Tech J, 27: 379–423 (1948)

    MATH  MathSciNet  Google Scholar 

  2. Weston J D. A note on the theory of communication. Philos Mag, 40: 449–453 (1939)

    MathSciNet  Google Scholar 

  3. Zayed A. Kramer’s sampling theorem for multidimensional signals and its relationship Lagrange-type interpolation. Multidimens Syst Signal Process, 3: 323–340 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Whittaker J M. Interpolatory Function Theory. Cambridge: Cambridge University Press, 1935

    Google Scholar 

  5. Levin B Ya. Lectures on entire functions. In: Translations of Mathematical Monographs 150. Providence: American Mathematical Society, 1996

    Google Scholar 

  6. Benedetto J J. Irregular sampling and frames. In: Wavelets: A Tutorial in Theory and Applications, Chui C K, ed. San Diego: Acadenic Press, 1992, 445–507

    Google Scholar 

  7. Butzer P L. A survey of the Whittaker-Shannon sampling theorem and some of its extension. J Math Res Exposition, 3: 185–212 (1983)

    MathSciNet  Google Scholar 

  8. Butzer P L, Higgins J R, Stens R L. Classical and approximate sampling theorems; studies in the Lp(R) and the uniform norm. J Approx Theory, 137: 250–263 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Higgins J R. Five short stories about the cardinal series. Bull Amer Math Soc, 121: 45–89 (1985)

    Article  MathSciNet  Google Scholar 

  10. Paley R, Wiener N. Fourier Transform in Complex Domain. In: Colloq Publ Vol. 19. Providence: America Mathematical Society, 1934

    Google Scholar 

  11. Sun W. Density of wavelet frames. Appl Comput Anal, 22: 264–272 (2007)

    Article  MATH  Google Scholar 

  12. Zayed A. A sampling theorem for signals band-limited to general domain in several dimensions. J Math Anal Appl, 187: 196–211 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Aldroubi A, Gröchenig K. Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Review, 43: 585–620 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hinsen G. Explicit irregular sampling formulas. J Comp Appl Math, 40: 177–198 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zayed A. Sampling in Hilbert space. Proc Amer Math Soc, 124: 3767–3776 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hinsen G. Irregular sampling of band-limited L p-functions. J Approx Theory, 72: 346–364 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kadec M I. The exact value of the Paley-Wiener constant. Soviet Math Dokl, 5: 559–561 (1964)

    Google Scholar 

  18. Fang G S. Whittaker-Kotelnikov-Shannnon sampling theorem and aliasing error. J Approx Theory, 85: 115–131 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nikolskii S M. Approximation of Functions of Several Variables and Imbedding Theorems. Berlin: Springer-Verlag, 1975

    Google Scholar 

  20. Parzen E. A simple proof and some extensions of sampling theorems. Tech. Rep. 7. Stanford: Stanford University, 1956

    Google Scholar 

  21. Wang J J, Fang G S. Multidimensional sampling theorem and estimate of aliasing error (in Chinese). Acta Math Appl Sin, 19: 481–488 (1996)

    MATH  MathSciNet  Google Scholar 

  22. Zygmund A. Trigonometric Polynomials. Vols. I, II. Cambridge: Cambridge University Press, 1959

    Google Scholar 

  23. Young R M. An Introduction to Non-harmonic Fourier Series. New York: Academic Press, 1980

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Engineering, Xihua University, Chengdu, 610039, China

    GuangGui Chen

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    GuangGui Chen & GenSun Fang

Authors
  1. GuangGui Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. GenSun Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to GuangGui Chen.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10671019), Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007) and Key Project of Technology Bureau of Sichuan Province (Grant No. 05JY029-138)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, G., Fang, G. Multivariate irregular sampling theorem. Sci. China Ser. A-Math. 52, 2469–2478 (2009). https://doi.org/10.1007/s11425-009-0139-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-009-0139-y

Keywords

MSC(2000)

Search

Navigation