Skip to main content Accessibility help
×
Home

Solvability of cardinal spline interpolation problems

  • T. N. T. Goodman (a1)

Synopsis

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

Copyright

References

Hide All
1de Boor, C.. Odd-degree spline interpolation at a biinfinite knot sequence. In Approximation Theory, pp.3053 (Schaback, R. and Scherer, K. eds). Lecture Notes in Mathematics 556 (Heidelberg: Springer, 1976).
2Gantmacher, F. R.. The Theory of Matrices, Vols I and II (New York: Chelsea, 1964).
3Goodman, T. N. T.. , Hermite-Birkhoffinterpolation by Hermite-Birkhoff splines. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 195201.
4Goodman, T. N. T. and Lee, S. L.. The Budan-Fourier theorem and Hermite-Birkhoff spline interpolation. Trans. Amer. Math. Soc. 271 (1982), 451467.
5Jetter, K., Lorentz, G. G. and Riemenschneider, S. D.. Birkhoff Interpolation. Encyclopedia of Mathematics and its Applications, Vol. 19 (London: Addison-Wesley, 1983).
6Lee, S. L. and Shanrfa, A.. Cardinal lacunary interpolation by g-splines I. The characteristic polynomials. J. Approx. Theory 16 (1976), 8596.
7Lee, S. L.. A class of lacunary interpolation problems by spline functions. Aequationes Math. 18 (1978), 6476.
8Lee, S. L., Micchelli, C. A., Sharma, A. and Smith, P. W.. Some properties of periodic B-spline collocation matrices. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 235246.
9Micchelli, C. A.. Oscillation matrices and cardinal spline interpolation. Studies in Spline Functions and Approximation Theory, pp. 163201 (New York: Academic Press, 1976).
10Schoenberg, I. J., Cardinal Spline Interpolation. Regional Conf. Series in Appl. Math. 12 (Philadelphia: SIAM, 1973).
11Subbotin, Ju. N.. On the relations between finite differences and the corresponding derivative. Proc. Steklov Inst. Mat. 78 (1965), 2442; Eng. transl. by Amer. Math. Soc. (1967), 23–42.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed