Skip to main content
Log in

Interpolating Generalized Shannon Sampling Operators, Their Norms and Approximation Properties

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

This paper deals with approximations of bounded, uniformly continuous on the whole real axis, functions by the interpolating Shannon sampling operators. These interpolating operators are defined by the kernels obtained by dilation from known ones, in particular, such as Rogosinski and Blackman-Harris kernels. The focus of our study is on estimates for operator norms and approximation orders via moduli of continuity.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. H. H. Albrecht, A family of cosine-sum windows for high resolution measurements, in IEEE International Conference on Acoustics, Speech and Signal Processing, Salt Lake City, Mai 2001, Salt Lake City 2001, pp. 3081–73084.

  2. R. B. Blackman and J. W. Tukey, The measurement of power spectra, Dover, New York, 1958.

    MATH  Google Scholar 

  3. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation (vol. 1), Birkhauser Verlag, Basel, Stuttgart, 1971.

  4. P. L. Butzer, W. Engels, S. Ries and R. L. Stens, The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM J. Appl. Math., 46, 299–323, 1986.

    Article  MathSciNet  Google Scholar 

  5. P. L. Butzer, G. Schmeisser and R. L. Stens, An introduction to sampling analysis, in Nonuniform Sampling, Theory and Practice, (F. Marvasti, ed.), Kluwer, New York, 2001, pp. 17–121.

    Chapter  Google Scholar 

  6. P. L. Butzer, W. Splettstößer and R. L. Stens, The sampling theorems and linear prediction in signal analysis, Jahresber. Deutsch. Math-Verein, 90, 1–70, 1988.

    MathSciNet  MATH  Google Scholar 

  7. F. J. Harris, On the use of windows for harmonic analysis, Proc. of the IEEE, 66, 51–83, 1978.

    Article  Google Scholar 

  8. J. R. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996.

    MATH  Google Scholar 

  9. A. J. Jerri, Recovering interpolation for generalized sampling sums of approximation theory. In: Proc. 1999 Intern. Workshop on Sampling Theory and Applications, Loen, Norway. Norwegian Univ. Sci. and Technology, 1999, 113–118.

  10. A. Kivinukk, Approximation by typical sampling series. In: Proc. 1999 Intern. Workshop on Sampling Theory and Applications, Loen, Norway. Norwegian Univ. Sci. and Technology, 1999, 161–166.

  11. A. Kivinukk and G. Tamberg, Subordination in generalized sampling series by Rogosinski-type sampling series, in Proc. 1997 Intern. Workshop on Sampling Theory and Applications, Aveiro, Portugal, 1997, Univ. Aveiro, 1997, pp. 397–402.

  12. A. Kivinukk and G. Tamberg, On sampling operators defined by the Hann window and some of their extensions Sampling Theory in Signal and Image Processing, 2, 235–258, 2003.

    MathSciNet  MATH  Google Scholar 

  13. A. Kivinukk and G. Tamberg, Blackman-type windows for sampling series, J. of Comp. Analysis and Applications, 7, 361–372, 2005.

    MathSciNet  MATH  Google Scholar 

  14. A. Kivinukk and G. Tamberg, On Blackman-Harris Windows for Shannon Sampling Series, Sampling Theory in Signal and Image Processing, 6, 87–108, 2007.

    MathSciNet  MATH  Google Scholar 

  15. R. Lasser and J. Obermaier, Characterization of Blackman kernels as approximate identities, Analysis, 22, 13–19, 2002.

    Article  MathSciNet  Google Scholar 

  16. H. D. Meikle, A New Twist to Fourier Transforms, Wiley-VCH, Berlin, 2004.

    Book  Google Scholar 

  17. S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin, 1975. (Orig. Russian ed. Moscow, 1969)

    Book  Google Scholar 

  18. R. L. Stens, Sampling with generalized kernels, in Sampling Theory in Fourier and Signal Analysis: Advanced Topics, (J.R. Higgins and R.L. Stens, eds.), Clarendon Press, Oxford, 1999.

    MATH  Google Scholar 

  19. A. F. Timan, Theory of Approximation of Functions of a Real Variable, MacMillan, New York, 1965. (Orig. Russian ed. Moscow, 1960).

    Google Scholar 

  20. K. Turkowski, Filters for common resampling tasks. In: Graphics Gems I, A. S. Glassner, ed., Academic Press, 1990, pp. 147–165.

    Chapter  Google Scholar 

Download references

Acknowledgments

This research was partially supported by the Estonian Sci. Foundation, grants 6943, 7033, and by the Estonian Min. of Educ. and Research, projects SF0132723s06, SF0140011s09. The second author is grateful to the RICAM (Austrian Academy of Sciences) for support.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andi Kivinukk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kivinukk, A., Tamberg, G. Interpolating Generalized Shannon Sampling Operators, Their Norms and Approximation Properties. STSIP 8, 77–95 (2009). https://doi.org/10.1007/BF03549509

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549509

Key words and phrases

2000 AMS Mathematics Subject Classification

Navigation