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Circuits, Systems, and Signal Processing

, Volume 35, Issue 2, pp 719–729 | Cite as

A New Method for Chebyshev Polynomial Interpolation Based on Cosine Transforms

  • Bing-Zhao LiEmail author
  • Yan-Li Zhang
  • Xian Wang
  • Qi-Yuan Cheng
Short Paper
  • 295 Downloads

Abstract

Interpolation plays an important role in the areas of signal processing and applied mathematics. Among the various interpolation methods, those related to Chebyshev polynomial interpolation have received much interest recently. In this paper, we propose a new interpolation method using a type I discrete cosine transform (type I DCT) and the nonuniform roots of the second type of Chebyshev polynomials. In this method, the interpolation coefficients are derived using the type I DCT of the Chebyshev nonuniform sampling points. Simulations show the correctness of the proposed method, and a comparison of the proposed method with existing methods is also discussed in detail.

Keywords

Chebyshev polynomial Chebyshev nonuniform sampling Polynomial interpolation Coefficients Discrete cosine transform Error analysis 

Notes

Acknowledgments

The authors would like to thank the academic editor and anonymous reviewers for their valuable comments and suggestions. The authors would also like to thank Prof. Huafei Sun and Dr. Didar of the Beijing Institute of Technology for many discussions about and proofreading of the manuscript.

References

  1. 1.
    B. Adcock, A.C. Hansen, A. Shadrin, A stability barrier for reconstructions from Fourier samples. SIAM J. Numer. Anal. 52(1), 125–139 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    J.I. Agbinya, Interpolation using the discrete cosine transform. Electron. Lett. 28(20), 1928 (1992)CrossRefGoogle Scholar
  3. 3.
    M. Berzins, Adaptive polynomial interpolation on evenly spaced meshes. SIAM Rev. 49(4), 604–627 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    R.L. Burden, J.D. Faires, Numerical Analysis (Brooks Cole, Boston, 2010)Google Scholar
  5. 5.
    E.W. Cheney, Approximation Theory III (Academic Press Inc., New York, 1980)zbMATHGoogle Scholar
  6. 6.
    A. Dutt, M. Gu, V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation. SIAM J. Numer. Anal. 33(5), 1689–1711 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Fax, I.B. Parker, Chebyshev Polynomials in Numerical Analysis (Oxford University Press, London, 1968)Google Scholar
  8. 8.
    K. Glashoff, K. Roleff, A new method for Chebyshev approximation of complex-valued functions. Math. Comput. 36(153), 233–239 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    M.A. Hernandez, Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3), 433–445 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    M.F. Huber, Chebyshev polynomial Kalman filter. Digit. Signal. Process. 23(5), 1620–1629 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C.P. Li, B.Z. Li, T.Z. Xu, Approximating bandlimited signals associated with the LCT domain from nonuniform samples at unknown locations. Signal Process. 92(7), 1658–1664 (2012)CrossRefGoogle Scholar
  12. 12.
    B.Z. Li, T.Z. Xu, Spectral analysis of sampled signals in the linear canonical transform domain. Math. Probl. Eng. 2012, 1–19 (2012)zbMATHGoogle Scholar
  13. 13.
    P.S. Malachivskyy, Y.V. Pizyur, N.V. Danchak, E.B. Orazov, Chebyshev approximation by exponential-power expression. Cybern. Syst. Anal. 49(6), 877–881 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Y. Mao, Z. Yu, Interpolation using the type I discrete cosine transform with greatly increased accuracy. Electron. Lett. 31(2), 85–86 (1995)CrossRefGoogle Scholar
  15. 15.
    I.I. Marks, J. Robert, Introduction to Shannon Sampling and Interpolation Theory (Springer, New York, 1991)zbMATHCrossRefGoogle Scholar
  16. 16.
    P.P. Massopust, Interpolation and Approximation with Splines and Fractals (Oxford University Press, London, 2010)zbMATHGoogle Scholar
  17. 17.
    V.E. Neagoe, A two-dimensional nonuniform sampling expansion model. Signal Process. 33(1), 1–21 (1993)zbMATHCrossRefGoogle Scholar
  18. 18.
    V.E. Neagoe, Chebyshev nonuniform sampling cascaded with the discrete cosine transform for optimum interpolation. IEEE Trans. Acoust. Speech Signal Process. 38(10), 1812–1815 (1990)zbMATHCrossRefGoogle Scholar
  19. 19.
    D. Potts, M. Tasche, Sparse polynomial interpolation in Chebyshev bases. Linear Algebra Appl. 441, 61–87 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Puschel, J.F. Moura, Algebric signal processing theory. ArXiv preprint. cs/0612077(2006)
  21. 21.
    M. Puschel, J.F. Moura, Algebraic signal processing theory: foundation and 1-D time. IEEE Trans. Signal Process. 56(8), 3572–3585 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Puschel, J.F. Moura, Algebraic signal processing theory: 1-D space. IEEE Trans. Signal Process. 56(8), 3586–3599 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory (Wiley, New York, 1990)zbMATHGoogle Scholar
  24. 24.
    L.A. Sakhnovich, Interpolation Theory and Its Applications (Springer, New York, 1997)zbMATHCrossRefGoogle Scholar
  25. 25.
    G. Strang, The discrete cosine transform. SIAM Rev. 41(1), 135–147 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    L.N. Trefethen, Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50(1), 67–87 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    L.N. Trefethen, Approximation Theory and Approximation Practice (Society for Industrial and Applied Mathematics, Philadelphia, 2013)zbMATHGoogle Scholar
  28. 28.
    H. Triebel, Interpolation Theory, Tunction Spaces, Differential Operators (Wiley-VCH, Johann Ambrosius Barth, Heidelberg, 1995)Google Scholar
  29. 29.
    Z. Wang, Interpolation using type I discrete cosine transform. Electron. lett. 26(15), 1170–1172 (1990)CrossRefGoogle Scholar
  30. 30.
    Z. Wang, Interpolation using the discrete cosine transform: reconsideration. Electron. Lett. 29(2), 198–200 (1993)CrossRefGoogle Scholar
  31. 31.
    Z. Wang, G.A. Jullien, W.C. Miller, Fast computation of Chebyshev optimal nonuniform interpolation. Proceedings of the 38th Midwest Symposium on Circuits and Systems. 1, 111–114 (1995)Google Scholar
  32. 32.
    Z. Wang, L. Wang, Interpolation using the fast discrete sine transform. Signal Process. 26(1), 131–137 (1992)zbMATHCrossRefGoogle Scholar
  33. 33.
    T.Z. Xu, B.Z. Li, Linear Canonical Transform and its Applications (Science Press, Beijing, 2013)Google Scholar
  34. 34.
    A. Zakhor, G. Alvstad, Two-dimensional polynomial interpolation from nonuniform samples. IEEE Trans. Signal Process. 40(1), 169–180 (1992)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Bing-Zhao Li
    • 1
    • 2
    Email author
  • Yan-Li Zhang
    • 1
    • 2
  • Xian Wang
    • 1
    • 2
  • Qi-Yuan Cheng
    • 1
  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  2. 2.Beijing Key Laboratory of Fractional Signals and SystemsBeijingChina

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