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Interpolatory subdivision schemes with infinite masks originated from splines

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Abstract

A generic technique for the construction of diversity of interpolatory subdivision schemes on the base of polynomial and discrete splines is presented in the paper. The devised schemes have rational symbols and infinite masks but they are competitive (regularity, speed of convergence, computational complexity) with the schemes that have finite masks. We prove exponential decay of basic limit functions of the schemes with rational symbols and establish conditions, which guaranty the convergence of such schemes on initial data of power growth.

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References

  1. M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover Publ. Inc., New York, 1972).

    Google Scholar 

  2. J.H. Ahlberg, E.N. Nilson and J.L. Walsh, The Theory of Splines and their Applications (Academic Press, New York, 1967).

    MATH  Google Scholar 

  3. A.Z. Averbuch, A.B. Pevnyi and V.A. Zheludev, Butterworth wavelet transforms derived from discrete interpolatory splines: Recursive implementation, Signal Process. 81 (2001) 2363–2382.

    Article  MATH  Google Scholar 

  4. A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana 12 (1996) 527–591.

    MATH  MathSciNet  Google Scholar 

  5. I. Daubechies and Y. Huang, A decay theorem for refinable functions, Appl. Math. Lett. 7 (1994) 1–4.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992).

    MATH  Google Scholar 

  7. G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx. 5 (1989) 49–68.

    Article  MATH  MathSciNet  Google Scholar 

  8. N. Dyn, J.A. Gregory and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991) 127–147.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. Dyn, Analysis of convergence and smoothness by the formalism of Laurent polynomials, in: Tutorials on Multiresolution in Geometric Modelling, eds. A. Iske, E. Quak and M.S. Floater (Springer, 2002) pp. 51–68.

  10. C. Herley and M. Vetterli, Wavelets and recursive filter banks, IEEE Trans. Signal Process. 41(12) (1993) 2536–2556.

    Article  MATH  Google Scholar 

  11. A.B. Pevnyi and V.A. Zheludev, On the interpolation by discrete splines with equidistant nodes, J. Approx. Theory 102 (2000) 286–301.

    Article  MATH  MathSciNet  Google Scholar 

  12. A.V. Oppenheim and R.W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, 1989).

    MATH  Google Scholar 

  13. M. Unser, A. Aldroubi and M. Eden, B-spline signal processing: Part II – Efficient design and applications, IEEE Trans. Signal Process. 41(2) (1993) 834–848.

    Article  MATH  Google Scholar 

  14. I.J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory 2 (1969) 167–206.

    Article  MATH  MathSciNet  Google Scholar 

  15. I.J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946) 45–99, 112–141.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. V.A. Zheludev, Local quasi-interpolating splines and Fourier transforms, Soviet Math. Dokl. 31 (1985) 573–577.

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Computer Science, Tel Aviv University, Israel

    Valery A. Zheludev

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  1. Valery A. Zheludev
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Communicated by T. Sauer

Mathematics subject classifications (2000)

65D17, 65D07, 93E11

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Zheludev, V.A. Interpolatory subdivision schemes with infinite masks originated from splines. Adv Comput Math 25, 475–506 (2006). https://doi.org/10.1007/s10444-004-4149-6

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  • DOI: https://doi.org/10.1007/s10444-004-4149-6

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