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Stability of spline approximation and the restoration of grid functions

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Abstract

For a functionf(x) ∈ H rω , defined on a uniform grid approximately, we propose a stable method for approximately restoring the function with the aid of polynomial splines. We derive uniform estimates for the deviations of the spline and its derivatives from the function and its derivatives.

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Literature cited

  1. S. G. Mikhlin and K. L. Smolitskii, Approximate Methods for Solution of Differential and Integral Equations, American Elsevier (1967).

  2. I.Babushka, M. Prager, and É. Vitasek, Numerical Processes in Differential Equations, Wiley (1969).

  3. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press (1967).

  4. V. A. Morozov, “On the problem of differentiation and some algorithms for approximating experimental functions,” in: Computational Methods and Programming, Vol. 24 [in Russian], Moscow (1970), pp. 46–62.

    Google Scholar 

  5. F. R. Loscalzo and T. D. Talbot, “Spline function approximation for solution of ordinary differential equations,” SIAM J. Numer. Anal.,4, No. 3, 433–445 (1967).

    Google Scholar 

  6. F. R. Loscalzo, “An introduction to the application of spline functions to initial value problems,” in: Theory and Applications of Spline Functions, Proc. of Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wisconsin, 1968, Academic Press (1969), pp. 37–64.

  7. R. S. Varga, “Error bounds for spline interpolation,” in: Approximations with Special Emphasis on Spline Functions. Proc. Sympos. Univ. of Wisconsin, Madison, Wisconsin, 1969, Academic Press (1969), pp. 367–386.

  8. B. K. Swartz and R. S. Varga, “Error bounds for spline and L-spline interpolation,” J. Approximation Theory,6, No. 1, 6–49 (1972).

    Google Scholar 

  9. Yu. N. Subbotin, “Diameter of the class W2L in L(0, 2π) and approximation by spline functions,” Matem. Zametki,7, No. 1, 43–52 (1970).

    Google Scholar 

  10. N. L. Patsko, “Approximation by splines on an interval,” Matem. Zametki,16, No. 3, 491–500 (1974).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Scientific Center, Academy of Sciences of the USSR, USSR

    I. A. Pakhnutov

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  1. I. A. Pakhnutov
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Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 537–544, October, 1974.

The author thanks Yu. N. Subbotin for his statement of the problem and a discussion of the results.

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Pakhnutov, I.A. Stability of spline approximation and the restoration of grid functions. Mathematical Notes of the Academy of Sciences of the USSR 16, 910–914 (1974). https://doi.org/10.1007/BF01104254

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  • DOI: https://doi.org/10.1007/BF01104254

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