On the relations between finite differences and derivatives of cardinal spline functions

  • Hennie ter Morsche
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 501)


Let m be a natural number and let Sm denote the class of cardinal spline functions of degree m. The object of this note is to establish a linear relationship between the 2m+2 quantities s(i+x), s(i+1+x),...,s(i+m+x), s(k)(i+y), s(k)(i+1+y),...,s(k)(i+m+y), where x,y ∈ [0,1], i=0,±1,±2,... s ∈ Sm and where s(k) denotes the k-th derivative of s (k=0,1,2,...,m−1). Using the shift operator E, we represent this relation in a simple form, involving the exponential Euler polynomials. The results are applied to cardinal spline interpolation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    BÖHMER, K., MEINARDUS, G. and SCHEMPP, W.: Spline-Funktionen. Mannheim-Wien-Zürich, B.I.-Wissenshaftsverlag 1974.zbMATHGoogle Scholar
  2. [2]
    BOOR de, C. and SCHOENBERG, I.J.: Cardinal Interpolation and Spline functions VIII. The Budan-Fourier theorem for splines and applications. MRC T.S.R. 1546, May 1975.Google Scholar
  3. [3]
    FYFE, D.J.: Linear Dependence Relations Connecting Equal Interval N—the Degree Splines and Their Derivatives. J. Inst. Maths. Applics. 7 (1971), 398–406.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    MEIR, A. and SHARMA, A.: Convergence of a Class of Interpolatory Splines. J. Approximation Theory 1 (1968), 243–250.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    NILSON, E.N.: Polynomial Splines and a Fundamental Eigenvalue Problem for Polynomials. J. Approximation Theory 6 (1972), 439–465.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    NÖRLUND, N.E.: Vorlesungen über Differenzenrechnung. Berlin, Springer, 1924.CrossRefzbMATHGoogle Scholar
  7. [7]
    SCHOENBERG, I.J.: Cardinal Spline Interpolation, CBMS Vol. 12, Philadelphia, SIAM 1973.CrossRefzbMATHGoogle Scholar
  8. [8]
    WEBER, H.: Lehrbuch der Algebra, Erster band. Braunschweig, Friedr. Vieweg & Sohn, 1912.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Hennie ter Morsche
    • 1
  1. 1.Department of MathematicsTechnological University EindhovenEindhovenThe Netherlands

Personalised recommendations