Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 154637, 9 pages
http://dx.doi.org/10.1155/2013/154637
Research Article

Uniform Bounds of Aliasing and Truncated Errors in Sampling Series of Functions from Anisotropic Besov Class

School of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received 1 May 2013; Accepted 11 June 2013

Academic Editor: Yiming Ying

Copyright © 2013 Peixin Ye and Yongjie Han. 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. C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379–423, 1948. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. L. Butzer, W. Engels, and U. Scheben, “Magnitude of the truncation error in sam-pling expansions of bandlimited signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 30, no. 6, pp. 906–912, 1982. View at Google Scholar
  3. A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, Fla, USA, 1993. View at MathSciNet
  4. S. D. Casey and D. F. Walnut, “Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms,” SIAM Review, vol. 36, no. 4, pp. 537–577, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. L. Butzer, G. Schmeisser, and R. L. Stens, “An introduction to sampling analysis,” in Nonuniform Sampling, Theory and Practice, F. Marvasti, Ed., pp. 17–121, Kluwer Academic, New York, NY, USA, 2001. View at Google Scholar · View at MathSciNet
  6. S. Smale and D.-X. Zhou, “Shannon sampling and function reconstruction from point values,” Bulletin of the American Mathematical Society, vol. 41, no. 3, pp. 279–305, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Smale and D.-X. Zhou, “Shannon sampling. II. Connections to learning theory,” Applied and Computational Harmonic Analysis, vol. 19, no. 3, pp. 285–302, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. P. L. Butzer and J. Lei, “Approximation of signals using measured sampled values and error analysis,” Communications in Applied Analysis, vol. 4, no. 2, pp. 245–255, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. X. Ye, “Error analysis for Shannon sampling series approximation with measured sampled values,” Research Journal of Applied Sciences, Engineering and Technology, vol. 5, no. 3, pp. 858–864, 2013. View at Google Scholar
  10. J. J. Wang and G. S. Fang, “A multidimensional sampling theorem and an estimate of the aliasing error,” Acta Mathematicae Applicatae Sinica, vol. 19, no. 4, pp. 481–488, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. M. Li, “Uniform bounds for sampling expansions,” Journal of Approximation Theory, vol. 93, no. 1, pp. 100–113, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Jingfan and F. Gensun, “On uniform truncation error bounds and aliasing error for multidimensional sampling expansion,” Sampling Theory in Signal and Image Processing, vol. 2, no. 2, pp. 103–115, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. P. L. Butzer and J. Lei, “Errors in truncated sampling series with measured sampled values for not-necessarily bandlimited functions,” Functiones et Approximatio, vol. 26, pp. 25–39, 1998. View at Google Scholar · View at MathSciNet
  14. H. D. Helms and J. B. Thomas, “Truncation error of sampling-theorem expansions,” Proceedings of The IRE, vol. 50, no. 2, pp. 179–184, 1962. View at Google Scholar · View at MathSciNet
  15. D. Jagerman, “Bounds for truncation error of the sampling expansion,” SIAM Journal on Applied Mathematics, vol. 14, no. 4, pp. 714–723, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. A. Micchelli, Y. Xu, and H. Zhang, “Optimal learning of bandlimited functions from localized sampling,” Journal of Complexity, vol. 25, no. 2, pp. 85–114, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Ya. Olenko and T. K. Pogány, “Universal truncation error upper bounds in sampling restoration,” Georgian Mathematical Journal, vol. 17, no. 4, pp. 765–786, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P.-X. Ye and Z.-H. Song, “Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion,” Applied Mathematics B, vol. 27, no. 4, pp. 412–418, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. P. X. Ye, B. H. Sheng, and X. H. Yuan, “Optimal order of truncation and aliasing errors for multi-dimensional whittaker-shannon sampling expansion,” International Journal of Wireless and Mobile Computing, vol. 5, no. 4, pp. 327–333, 2012. View at Google Scholar
  20. S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, New York, NY, USA, 1975. View at MathSciNet
  21. Y. Jiang and Y. Liu, “Average widths and optimal recovery of multivariate Besov classes in Lp(Rd),” Journal of Approximation Theory, vol. 102, no. 1, pp. 155–170, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. A. Micchelli, Y. S. Xu, and P. X. Ye, “Cucker-Smale learning theory in Besov spaces,” in Advances in LearnIng Theory: Methods, Models and Applications, J. Suykens, G. Horvath, S. Basu et al., Eds., pp. 47–68, IOS Press, Amsterdam, The Netherlands, 2003. View at Google Scholar
  23. R. P. Boas, Jr., Entire Functions, Academic Press, New York, NY, USA, 1954. View at MathSciNet
  24. G. Fang, F. J. Hickernell, and H. Li, “Approximation on anisotropic Besov classes with mixed norms by standard information,” Journal of Complexity, vol. 21, no. 3, pp. 294–313, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. H. G. Burchard and J. Lei, “Coordinate order of approximation by functional-based approximation operators,” Journal of Approximation Theory, vol. 82, no. 2, pp. 240–256, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet