Cardinal Splines

  • Carl de Boor
Part of the Contemporary Mathematicians book series (CM)


Splines came out of the barrel of a cannon. Papers on what we now call splines had appeared before Schoenberg’s basic spline paper [31*] on techniques for interpolation and smoothing of ballistic tables. Two of the most interesting ones, [Ea] and [QC], even share with [31*] the subject matter, namely cardinal spline interpolation, and approach it in the same way, namely with the aid of the Fourier transform. [31*] is nevertheless the foundation of spline theory since it is the first paper which treats smooth piecewise polynomials not just as useful auxiliary constructs but as objects deserving of study in their own right, in token of which it provided them with a name, splines. More than that, [31*] bases its analysis of splines on that most useful of spline functions, the B-spline, which is introduced there in terms of its Fourier transform.


Spline Space Spline Interpolant Cardinal Spline Ballistic Table Spline Theory 
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.


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  1. [B]
    C. de Boor (1976), Splines as Linear Combinations of B-Splines, in Approximation Theory II, G. G. Lorentz, C. K. Chui and L. L. Schumaker, eds., Academic Press, 1-47.Google Scholar
  2. [BH]
    C. de Boor and K. Höllig (1982/83), B-splines from Parallelepipeds, J. d’Anal. Math. 42, 99–115.CrossRefGoogle Scholar
  3. [DM]
    W. Dahmen and C.A. Micchelli, Some results on box splines, Bull. Amer. Math. Soc. 11 (1984), 147–150.CrossRefGoogle Scholar
  4. [Ea]
    A. Eagle (1928), On the relation between Fourier constants of a periodic function and the coefficients determined by harmonic analysis, Phil.Mag. 5, 113–132.Google Scholar
  5. [QC]
    W. Quade & L. Collatz (1938), Zur Interpolationstheorie der reellen periodischen Funktionen, Akad.Wiss., Math.-Phys. Klasse 30, 383–429.Google Scholar
  6. [SF]
    G. Strang and G. Fix (1973), A Fourier Analysis of the Finite Element Variational Method, C.I.M.E., II Ciclo 1971, in Constructive Aspects of Functional Analysis, G. Geymonat ed., 793-840.Google Scholar

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Carl de Boor
    • 1
  1. 1.University of WisconsinMadisonUSA

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