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Interpolation by splines satisfying mixed boundary conditions

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Abstract

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen−k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.

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References

  1. D. Ferguson,The question of uniqueness for G. D. Birkhoff interpolation problems, J. Approximation Theory2 (1969), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  2. F. R. Gantmacher and M. G. Krein,Oscillatory matrices and kernels and small vibrations of mechanical systems, 2nd ed., Akademie Verlag Berlin, 1960.

    Google Scholar 

  3. S. Karlin,Total positivity, Vol. I, Stanford University Press, Stanford, California, 1968.

    MATH  Google Scholar 

  4. S. Karlin and J. W. Lee,Periodic boundary value problems with cyclic totally positive Green’s functions with applications to periodic spline theory, J. Differential Equations8 (1970), 374–396.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Karlin,Total positivity, interpolation by splines and Green’s functions of differential operators, J. Approximation Theory4 (1971), 91–112.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Karlin and C. Micchelli,The fundamental theorem of algebra for monosplines satisfying boundary conditions, Israel J. Math.11 (1972), 405–451.

    MATH  MathSciNet  Google Scholar 

  7. A. A. Melkman,The Budan-Fourier theorem for splines, Israel J. Math.19 (1974), 256–263.

    MathSciNet  Google Scholar 

  8. G. Polya,Bemerkungen zur Interpolation und zur Näherungs theorie der Balkenbiegung, Z. Angew. Math. Mech.11 (1931), 445–449.

    Article  MATH  Google Scholar 

  9. J. R. Rice,Characterization of Chebyshev approximation by splines, SIAM J. Numer. Anal.4 (1967), 557–565.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. J. Schoenberg and A. Whitney,On Polya frequency functions III: The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc.74 (1953), 246–259.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Applied Science, The Hebrew University of Jerusalem, Jerusalem, Israel

    Avraham A. Melkman

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  1. Avraham A. Melkman
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Melkman, A.A. Interpolation by splines satisfying mixed boundary conditions. Israel J. Math. 19, 369–381 (1974). https://doi.org/10.1007/BF02757500

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  • DOI: https://doi.org/10.1007/BF02757500

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