Skip to main content
SpringerLink
Log in

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Bogle, M.G.V., Hearst, J.E., Jones, V.F.R., Stoilov, L.: Lissajous knots. J. Knot Theory Ramifications 3(2), 121–140 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bos, L., Caliari, M., De Marchi, S., Vianello, M.: A numerical study of the Xu polynomial interpolation formula in two variables. Computing 76(3), 311–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bos, L., Caliari, M., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143(1), 15–25 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bos, L., De Marchi, S., Vianello, M.: On the Lebesgue constant for the Xu interpolation formula. J. Approx. Theory 141(2), 134–141 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bos, L., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the ideal theory approach. Numer. Math. 108(1), 43–57 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caliari, M., De Marchi, S., Sommariva, A., Vianello, M.: Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave. Numer. Algorit. 56(1), 45–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caliari, M., De Marchi, S., Vianello, M.: Bivariate polynomial interpolation on the square at new nodal sets. Appl. Math. Comput. 165(2), 261–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caliari, M., De Marchi, S., Vianello, M.: Bivariate Lagrange interpolation at the Padua points: computational aspects. J. Comput. Appl. Math. 221(2), 284–292 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caliari, M., Vianello, M., De Marchi, S., Montagna, R.: Hyper2d: a numerical code for hyperinterpolation on rectangles. Appl. Math. Comput. 183(2), 1138–1147 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cools, R.: Constructing cubature formulae: the science behind the art. Acta Numerica 6, 1–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cools, R., Poppe, K.: Chebyshev lattices, a unifying framework for cubature with Chebyshev weight function. BIT 51(2), 275–288 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  13. Franke, R.: Scattered data interpolation: Tests of some methods. Math. Comp. 38(157), 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  14. Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation. J. Comput. Appl. Math. 122(1–2), 23–35 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12(4), 377–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gleich, B., Weizenecker, J.: Tomographic imaging using the nonlinear response of magnetic particles. Nature 435(7046), 1214–1217 (2005)

    Article  Google Scholar 

  17. Grüttner, M., Knopp, T., Franke, J., Heidenreich, M., Rahmer, J., Halkola, A., Kaethner, C., Borgert, J., Buzug, T.M.: On the formulation of the image reconstruction problem in magnetic particle imaging. Biomed. Tech. Biomed. Eng. 58(6), 583–591 (2013)

    Google Scholar 

  18. Harris, L.A.: Bivariate Lagrange interpolation at the Geronimus nodes. Contemp. Math. 591, 135–147 (2013)

    Article  MATH  Google Scholar 

  19. Kaethner, C., Ahlborg, M., Bringout, G., Weber, M., Buzug, T.M.: Axially elongated field-free point data acquisition in magnetic particle imaging. IEEE Trans. Med. Imag. 34(2), 381–387 (2015)

  20. Knopp, T., Biederer, S., Sattel, T.F., Weizenecker, J., Gleich, B., Borgert, J., Buzug, T.M.: Trajectory analysis for magnetic particle imaging. Phys. Med. Biol. 54(2), 385–3971 (2009)

    Article  Google Scholar 

  21. Lamm, C.: There are infinitely many Lissajous knots. Manuscr. Math. 93(1), 29–37 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25, 185–200 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Morrow, C.R., Patterson, T.N.L.: Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal. 15, 953–976 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Renka, R.J., Brown, R.: Algorithm 792: accuracy tests of acm algorithms for interpolation of scattered data in the plane. ACM Trans. Math. Softw. 25(1), 78–94 (1999)

    Article  MATH  Google Scholar 

  25. Xu, Y.: Lagrange interpolation on Chebyshev points of two variables. J. Approx. Theory 87(2), 220–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zygmund, A.: Trigonometric Series, Vol I & II combined, 3rd edn. Cambridge University Press, Cambridge Mathematical Library, Cambridge (2002)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the financial support of the German Federal Ministry of Education and Research (BMBF, Grant number 13N11090), the German Research Foundation (DFG, Grant number BU 1436/9-1 and ER 777/1-1), the European Union and the State Schleswig-Holstein (EFRE, grant number 122-10-004).

Author information

Authors and Affiliations

  1. Institute of Mathematics, Universität zu Lübeck, Ratzeburger Allee 160, 23562, Lübeck, Germany

    Wolfgang Erb

  2. Institute of Medical Engineering, Universität zu Lübeck, Ratzeburger Allee 160, 23562, Lübeck, Germany

    Christian Kaethner, Mandy Ahlborg & Thorsten M. Buzug

Authors
  1. Wolfgang Erb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Kaethner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mandy Ahlborg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thorsten M. Buzug
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Kaethner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Erb, W., Kaethner, C., Ahlborg, M. et al. Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves. Numer. Math. 133, 685–705 (2016). https://doi.org/10.1007/s00211-015-0762-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0762-1

Mathematics Subject Classification

Search

Navigation