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Hermite–Birkhoff interpolation by Hermite–Birkhoff splines

Published online by Cambridge University Press:  14 November 2011

T. N. T. Goodman
T. N. T. Goodman
Affiliation:
Department of Mathematics, University of Dundee
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Synopsis

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E. Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F. We first find necessary conditions on (E, F) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on (E, F) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 195 - 201
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Jetter, K.. Birkhoff interpolation by spline s. In. Approximation Theory II, pp. 405–410 (Lorentz, G. G., Chui, C. K. and Schumaker, L. L., Eds) (New York: Academic, 1976).Google Scholar
2Karlin, S. and Karon, J.. On Hermite–Birkhoff interpolation. J. Approximation. Theory 6 (1972), 90–114.CrossRefGoogle Scholar
3Melkman, A. A. The Budan–Fourier theorem for splines. Israel J. Math. 19 (1974), 256–263.CrossRefGoogle Scholar
4Melkman, A. A.. Hermite–Birkhoff interpolation by splines. J. Approximation. Theory 19 (1977), 259–279.CrossRefGoogle Scholar
5Pence, D. D.. Hermite–Birkhoff interpolation and monotone approximation by splines. J. Approximation.Theory 25 (1979), 248–257.Google Scholar
6Sharma, A.. Some poised and non-poised problems of interpolation. SIAM Rev. 14 (1972), 129–151.Google Scholar