Spline Interpolation and Best Quadrature Formulae

  • I. J. Schoenberg
Part of the Contemporary Mathematicians book series (CM)

Abstract

The spline interpolation formula. A spline function S(x), of degree k(≧0), having the knots
$$x_0 < x_1 < \cdots < x_n ,$$
(1)
is by definition a function of the class C k−1 which reduces to a polynomial of degree not exceeding k in each of the n+2 intervals in which the points (1) divide the real axis. The function S(x) is seen to depend linearly on n+k +1 parameters. In [5, Theorem 2, p. 258] are given the precise conditions under which we can interpolate uniquely by S(x) arbitrarily given ordinates at n+k +1 points on the real axis.

Keywords

Quadrature Formula Spline Function Spline Interpolation Minimal Property Mechanical Quadrature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. de Boor, Best approximation properties of spline functions of odd degree, J. Math. Mech. 12 (1963), 747–749.Google Scholar
  2. 2.
    J. C. Holladay, Smoothest curve approximation, Math. Tables Aids Comput. 11 (1957), 233–243.CrossRefGoogle Scholar
  3. 3.
    G. Peano, Residuo in formulas de quadratura, Mathesis 34 (1914), 1–10.Google Scholar
  4. 4.
    A. Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949), 80–91.CrossRefGoogle Scholar
  5. 5.
    I. J. Schoenberg and Anne Whitney, On Polya frequency functions. III. The positivity of translation determinants with an application to the interpolation by spline curves, Trans. Amer. Math. Soc. 74 (1953), 246–259.Google Scholar
  6. 6.
    I. J. Schoenberg, Spline functions, convex curves and mechanical quadratures, Bull. Amer. Math. Soc. 64 (1958), pp. 352–357.CrossRefGoogle Scholar
  7. 7.
    —, On interpolation by spline functions and its minimal properties, Proceedings of the Conference on Approximation, Oberwolfach, Germany, August 4–10, 1963. (to appear)Google Scholar
  8. 8.
    J. L. Walsh, J. H., Ahlberg and E. N. Nilson, Best approximation properties of the spline fit, J. Math. Mech. 11 (1962), 225–234.Google Scholar
  9. 9.
    J. H., Ahlberg and E. N. Nilson —, Best approximation and convergence properties of higher-order spline fits, Abstract 63t-103, Notices Amer. Math. Soc. 10 (1963), 202.Google Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • I. J. Schoenberg
    • 1
  1. 1.Institute for Advanced StudyUniversity of PennsylvaniaUSA

Personalised recommendations