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Reconstruction of Splines from Nonuniform Samples

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Abstract

In this paper, we study the reconstruction of spline functions from their nonuniform samples. We investigate the existence and uniqueness of the solution of the following problem: for given data {(xn, yn): n ∈ ℤ}, find a cardinal spline f(x), of a given degree, satisfying yn = f(xn), n ∈ ℤ. Several necessary and/or sufficient conditions for the existence and uniqueness of the solution of the problem are derived. Finally, an example and some applications are presented to illustrate the main results.

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References

  1. Aldroubi, A., Gröchenig, K.: Beurling–Landau-type theorems for non-uniform sampling in shift invariant spline spaces. J. Fourier Anal. Appl., 6, 93–103 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldroubi, A., Gröhenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev., 43, 585–620 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aldroubi, A., Davis, J., Krishtal, I.: Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off. J. Fourier Anal. Appl., 21, 11–31 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Y., Cheng, C., Sun, Q., et al.: Phase retrieval of real-valued signals in a shift-invariant space. ArXiv:1603.01592

  5. Daubechies, I.: Ten Lectures on Wavelets, SIAM, Philadelphia, 1992

    Book  MATH  Google Scholar 

  6. Folland, B.: Real Analysis, John Wiley, New York, 1984

    MATH  Google Scholar 

  7. Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl., 10, 105–132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huo, H., Sun, W.: Average sampling theorem. Sci. China Ser. A, 45, 1403–1422 (2015)

    Google Scholar 

  9. Jia, R., Micchelli, C.: Using the refinement equations for the construction of pre-wavelets, ii. powers of two, in Curves and Surfaces (P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds.), Academic Press, Boston, MA, 209–246 (1991)

  10. Jia, R.: Shift-invariant spaces on the real line. Proc. Amer. Math. Soc., 125, 785–793 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Katznelson, Y.: An Introduction to Harmonic Analysis, John Wiley, New York, 1968

    MATH  Google Scholar 

  12. Landau, H.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math., 117, 37–52 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  13. Subbotin, J. N.: On the relations between finite differences and the corresponding derivatives. Proc. Steklov Inst. Math., 78, 24–42 (1965)

    Google Scholar 

  14. Schoenberg, I. J.: Cardinal interpolation and spline functions: II. interpolation of data of power growth. J. Approx. Theory, 6, 404–420 (1972)

    Article  MATH  Google Scholar 

  15. Schoenberg, I. J.: Cardinal Spline Interpolation, SIAM, Wisconsin, 1973

    Book  MATH  Google Scholar 

  16. Sun, W., Zhou, X.: On the sampling for wavelet subspaces. J. Fourier Anal. Appl., 5, 347–354 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun, W., Zhou, X.: Average sampling in spline subspaces. Appl. Math. Letters, 15, 233–237 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sun, W., Zhou, X.: Reconstruction of functions in spline subspaces from local averages. Proc. Amer. Math. Soc., 131, 2561–2571 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, W., Zhou, X.: Characterization of local sampling sequences for spline subspaces. Adv. Comput. Math., 30, 153–175 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xian, J., Luo, S., Lim, W.: Weighted sampling and signal reconstruction in spline subspaces. Signal Process., 86, 331–340 (2006)

    Article  MATH  Google Scholar 

  21. Xian, X., Li, S.: Sampling set conditions in weighted multiply generated shift-invariant spaces and their applications. Appl. Comput. Harmon. Anal., 23, 171–180 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xian, X., Li, S.: General A-P iterative algorithm in shift-invariant spaces. Acta Math. Sin., Engl. Ser., 25, 545–552 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Q., Liu, B., Li, R.: Dynamical sampling in multiply generated shift-invariant spaces. Appl. Anal., 96, 760–770 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Science, Tianjin University of Technology, Tianjin, 300384, P. R. China

    Qing Yue Zhang

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Wen Chang Sun

Authors
  1. Qing Yue Zhang
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  2. Wen Chang Sun
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Correspondence to Qing Yue Zhang.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11525104, 11531013, 11401435 and 11326094), the Fundamental Research Funds for the Central Universities and the Program for Visiting Scholars at the Chern Institute of Mathematics

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Zhang, Q.Y., Sun, W.C. Reconstruction of Splines from Nonuniform Samples. Acta. Math. Sin.-English Ser. 35, 245–256 (2019). https://doi.org/10.1007/s10114-018-7531-x

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  • DOI: https://doi.org/10.1007/s10114-018-7531-x

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