Skip to main content
SpringerLink
Log in

Optimal shift parameters for periodic spline interpolation

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Using the exponential Euler spline, restricted on the unit circle, we sketch a unified approach to the periodic spline interpolation with shifted interpolation nodes. Mainly we are interested in the optimal choice of the shift parameter τ such that the corresponding interpolatory matrix possesses minimal condition or such that the related interpolation operator has minimal norm. We show that τ=0 is optimal in both cases. This improves known results of Merz, Reimer-Siepmann and Richards.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abramowitz and J.A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1965).

    Google Scholar 

  2. P.J. Davis,Circulant Matrices (Wiley, New York, 1979).

    Google Scholar 

  3. K. Jetter, S.D. Riemenschneider and N. Sivakumar, Schoenberg's exponential Euler spline curves, Proc. Roy. Soc. Edinburgh 118A (1991) 21–33.

    Google Scholar 

  4. M. Marsden and R. Mureika, Cardinal spline interpolation inL 2, Illinois J. Math. 19 (1975) 145–147.

    Google Scholar 

  5. M.J. Marsden, F.B. Richards and S.D. Riemenschneider, Cardinal spline interpolation operators onl p data, Indiana Univ. Math. J. 24 (1975) 677–689.

    Article  Google Scholar 

  6. M.J. Marsden, F.B. Richards and S.D. Riemenschneider, Erratum to “Cardinal spline interpolation operators onl p data”, Indiana Univ. Math. J. 25 (1976) 919.

    Article  Google Scholar 

  7. G. Merz, Interpolation mit periodischen Spline-Funktionen I, J. Approx. Theory 30 (1980) 11–19.

    Article  Google Scholar 

  8. G. Merz, Interpolation mit periodischen Spline-Funktionen III, J. Approx. Theory 34 (1982) 226–236.

    Article  Google Scholar 

  9. H. ter Morsche, On the existence and convergence of interpolating periodic spline functions of arbitrary degree, in:Spline-Funktionen, eds. K. Böhmer, G. Meinardus and W. Schempp (Bibliographisches Institut, Mannheim, 1974) pp. 197–214.

    Google Scholar 

  10. M. Reimer and D. Siepmann, An elementary algebraic representation of polynomial spline interpolants for equidistant lattices and its condition, Numer. Math. 49 (1986) 55–65.

    Article  Google Scholar 

  11. F.B. Richards, Best bounds for the uniform periodic spline interpolation operator, J. Approx. Theory 7 (1973) 302–317.

    Article  Google Scholar 

  12. F.B. Richards, Uniform spline operators inL 2, Illinois J. Math. 18 (1974) 512–521.

    Google Scholar 

  13. I.J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory 2 (1969) 167–206.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Rostock, D-18051, Rostock, Germany

    Gerlind Plonka

Authors
  1. Gerlind Plonka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by C.A. Micchelli

Rights and permissions

Reprints and permissions

About this article

Cite this article

Plonka, G. Optimal shift parameters for periodic spline interpolation. Numer Algor 6, 297–316 (1994). https://doi.org/10.1007/BF02142676

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02142676

Keywords

Subject classification

Search

Navigation