### Abstract

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.

Original language | English |
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Title of host publication | Spline Functions (Proceedings International Symposium, Karlsruhe, Germany, 1975) |

Editors | K. Boehmer, G. Meinardus, W. Schempp |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 210-219 |

Number of pages | 10 |

ISBN (Electronic) | 978-3-540-38073-3 |

ISBN (Print) | 978-3-540-07543-1 |

DOIs | |

Publication status | Published - 1976 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 501 |

ISSN (Print) | 0075-8434 |

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## Cite this

Morsche, ter, H. G. (1976). On the relations between finite differences and derivatives of cardinal spline functions. In K. Boehmer, G. Meinardus, & W. Schempp (Eds.),

*Spline Functions (Proceedings International Symposium, Karlsruhe, Germany, 1975)*(pp. 210-219). (Lecture Notes in Mathematics; Vol. 501). Berlin: Springer. https://doi.org/10.1007/BFb0079749