Minimal Properties of Periodic Box-Spline Interpolation on a Three Direction Mesh

  • Joachim Stöckler
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 90)


Let ø be a piecewise continuous complex valued function on ℝ s , s ≥ 1, which is 2π-periodic in each coordinate direction. Given a “meshsize” h = 2π/N, N ≥ 1, let F := F h := h Z s ∩ [0, 2π) s and
$$S():={{S}_{h}}():=\left\{ \sum\limits_{j\in \text{F}}{{{a}_{j}}(.-j)\left| {{a}_{j}}\in \mathbb{C} \right.} \right\}$$


Discrete Fourier Transform Fourier Coefficient Interpolation Problem Attenuation Factor Coordinate Direction 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlberg, J. H., Nilson, E. N., Walsh, J. N. (1965) The Theory of Splines and their Applications ( Academic Press, New York-London).Google Scholar
  2. 2.
    de Boor, C., Höllig, K., Riemenschneider, S. D. (1985) Bivariate cardinal interpolation by splines on a three direction mesh. Illinois J. Math. 29, 533–566.Google Scholar
  3. 3.
    Delvos, F.-J. (1987) Optimal periodic interpolation in the mean. Preprint.Google Scholar
  4. 4.
    Ehlich, H. (1966) Untersuchungen zur numerischen Fourieranalyse. Math. Z. 91, 380–420.CrossRefGoogle Scholar
  5. 5.
    Golomb, M. (1968) Approximation by periodic spline interpolants on uniform meshes. J. Approximation Theory 1, 26–65.CrossRefGoogle Scholar
  6. 6.
    Gutknecht, M. H. (1987) Attenuation factors in multivariate Fourier analysis.- Numer. Math. 51, 615–629.Google Scholar
  7. 7.
    terMorsche, H. (1987) Attenuation factors and multivariate periodic spline interpolation. In: Topics in Multivariate Approximation, C. Chui, L. Schumaker, F. Utreras, eds. ( Academic Press, New York ), 165–174.Google Scholar
  8. 8.
    Sard, A. (1967/68) Optimal approximation, J. Funct. Anal. 1, 222–244; 2, 368–369.Google Scholar
  9. 9.
    Stöckler, J. (1988) Interpolation mit mehrdimensionalen Bernoulli-Splines und periodischen Box-Splines. Dissertation, Duisburg.Google Scholar
  10. 10.
    Stöckler, J. (1989) On minimum norm interpolation by multivariate Bernoulli splines. To appear in: Approximation Theory VI, C. Chui, L. Schumaker, J. Ward, eds. ( Academic Press, New York).Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • Joachim Stöckler
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of DuisburgWest-Germany
  2. 2.Fachbereich MathematikUniversität DuisburgDuisburg 1West-Germany

Personalised recommendations