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Approximation by Discrete GB-Splines

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Abstract

This paper addresses the definition and the study of discrete generalized splines. Discrete generalized splines are continuous piecewise defined functions which meet some smoothness conditions for the first and second divided differences at the knots. They provide a generalization both of smooth generalized splines and of the classical discrete cubic splines. Completely general configurations for steps in divided differences are considered. Direct algorithms are proposed for constructing discrete generalized splines and discrete generalized B-splines (discrete GB-splines for short). Explicit formulae and recurrence relations are obtained for discrete GB-splines. Properties of discrete GB-splines and their series are studied. It is shown that discrete GB-splines form weak Chebyshev systems and that series of discrete GB-splines have a variation diminishing property.

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References

  1. H. Akima, A new method of interpolation and smooth curve fitting based on local procedures, J. Assoc. Comput. Mech. 17 (1970) 589–602.

    Google Scholar 

  2. P.H. Astor and C.S. Duris, Discrete L-splines, Numer. Math. 22 (1974) 393–402.

    Google Scholar 

  3. J.C. Clement, Convexity-preserving piecewise rational cubic interpolation, SIAM J. Numer. Anal. 27 (1990) 1016–1023.

    Google Scholar 

  4. E. Cohen, T. Lyche and R. Riesenfeld, Discrete B-splines and subdivision techniques in computer aided geometric design and computer graphics, Comput. Graph. Image Process. 14 (1980) 87–111.

    Google Scholar 

  5. P. Costantini, B.I. Kvasov and C. Manni, On discrete hyperbolic tension splines, Adv. Comput. Math. 11 (1999) 331–354.

    Google Scholar 

  6. W. Dahmen and Ch.A. Micchelli, Algebraic properties of discrete box splines, Constr. Approx. 3 (1987) 209–221.

    Google Scholar 

  7. C. de Boor, Splines as linear combinations of B-splines: A survey, in: Approximation Theory II, eds. G.G. Lorentz, C.K. Chui and L.L. Schumaker (Academic Press, New York, 1976) pp. 1–47.

    Google Scholar 

  8. C. de Boor and A. Pinkus, Backward error analysis for totally positive linear systems, Numer. Math. 27 (1977) 485–490.

    Google Scholar 

  9. H.P. Dikshit and P. Powar, Discrete cubic spline interpolation, Numer. Math. 40 (1982) 71–78.

    Google Scholar 

  10. H.P. Dikshit and S.S. Rana, Discrete cubic spline interpolation over a nonuniform mesh, Rocky Mountain J. Math. 17 (1987) 709–718.

    Google Scholar 

  11. S. Karlin, Total Positivity, Vol. 1 (Stanford Univ. Press, Stanford, CA, 1968).

    Google Scholar 

  12. P.E. Koch and T. Lyche, Exponential B-splines in tension, in: Approximation Theory VI: Proc. of the 6th Internat. Symposium on Approximation Theory, Vol. II, eds. C.K. Chui, L.L. Schumaker and J.D. Ward (Academic Press, Boston, 1989) pp. 361–364.

    Google Scholar 

  13. B.I. Kvasov, Local bases for generalized cubic splines, Russian J. Numer. Anal. Math. Modelling 10 (1995) 49–80.

    Google Scholar 

  14. B.I. Kvasov, Shape preserving spline approximation via local algorithms, in: Advanced Topics in Multivariate Approximation, eds. F. Fontanella, K. Jetter and P.J. Laurent (World Scientific, Singapore, 1996) pp. 181–196.

    Google Scholar 

  15. B.I. Kvasov, GB-splines and their properties, Ann. Numer. Math. 3 (1996) 139–149.

    Google Scholar 

  16. B.I. Kvasov, Algorithms for shape preserving local approximation with automatic selection of tension parameters, Comput. Aided Geom. Design 17 (2000) 17–37.

    Google Scholar 

  17. B.I. Kvasov, Methods of Shape-Preserving Spline Approximation (World Scientific, Singapore, 2000). 188 B.I. Kvasov / Approximation by discrete GB-splines

    Google Scholar 

  18. T. Lyche, Discrete polynomial spline approximation methods, Thesis, University of Texas, Austin (1975). For a summary, see Spline Functions, eds. K. Böhmer, G. Meinardus and W. Schimpp, Karlsruhe, 1975, Lectures Notes in Mathematics, Vol. 501 (Springer, Berlin, 1976) pp. 144–176.

    Google Scholar 

  19. T. Lyche, Discrete cubic spline interpolation, BIT 16 (1976) 281–290.

    Google Scholar 

  20. M.A. Malcolm, On computation of nonlinear spline functions, SIAM J. Numer. Anal. 14 (1977) 254-282.

    Google Scholar 

  21. O.L. Mangasarian and L.L. Schumaker, Discrete splines via mathematical programming, SIAM J. Control 9 (1971) 174–183.

    Google Scholar 

  22. O.L. Mangasarian and L.L. Schumaker, Best summation formulae and discrete spline, SIAM J. Numer. Anal. 10 (1973) 448–459.

    Google Scholar 

  23. B.J. McCartin, Computation of exponential splines, SIAMJ. Sci. Statist. Comput. 11 (1990) 242–262.

    Google Scholar 

  24. A.A. Melkman, Another proof of the total positivity of the discrete spline collocation matrix, J. Approx. Theory 84 (1996) 247–264.

    Google Scholar 

  25. K.M. Mørken, On total positivity of the discrete spline collocation matrix, J. Approx. Theory 84 (1996) 247–264.

    Google Scholar 

  26. S.S. Rana, Convergence of a class of deficient interpolatory splines, Rocky Mountain J. Math. 14 (1988) 825–831.

    Google Scholar 

  27. S.S. Rana and Y.P. Dubey, Local behaviour of the deficient discrete cubic spline interpolator, J. Approx. Theory 86 (1996) 120–127.

    Google Scholar 

  28. R. Renka, Interpolatory tension splines with automatic selection of tension factors, SIAM J. Sci. Statist. Comput. 8 (1987) 393–415.

    Google Scholar 

  29. P. Rentrop, An algorithm for the computation of exponential splines, Numer. Math. 35 (1980) 81–93.

    Google Scholar 

  30. P. Rentrop and U. Wever, Computational strategies for the tension parameters of the exponential spline, in: Lecture Notes in Control and Information Science, Vol. 95 (1987) pp. 122–134.

    Google Scholar 

  31. A. Ron, Exponential box splines, Constructive Approx. 4 (1988) 357–378.

    Google Scholar 

  32. N.S. Sapidis and P.D. Kaklis, An algorithm for constructing convexity and monotonicity-preserving splines in tension, Comput. Aided Geom. Design 5 (1988) 127–137.

    Google Scholar 

  33. N.S. Sapidis, P.D. Kaklis and T.A. Loukakis, A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation, Numer. Math. 54 (1988) 179–192.

    Google Scholar 

  34. L.L. Schumaker, Constructive aspects of discrete polynomial spline functions, in: Approximation Theory, ed. G.G. Lorentz (Academic Press, New York, 1973) pp. 469–476.

    Google Scholar 

  35. L.L. Schumaker, Spline Functions: Basic Theory (Wiley, New York, 1981).

    Google Scholar 

  36. A.F. Shampine, R.C. Allen, Jr. and S. Pruess, Fundamentals of Numerical Computing (Wiley, New York, 1997).

    Google Scholar 

  37. Yu.S. Zav'yalov, B.I. Kvasov and V.L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) (in Russian).

    Google Scholar 

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  1. School of Mathematics, Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima, 30000, Thailand

    Boris I. Kvasov

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  1. Boris I. Kvasov
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Kvasov, B.I. Approximation by Discrete GB-Splines. Numerical Algorithms 27, 169–188 (2001). https://doi.org/10.1023/A:1011818621589

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