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Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh

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Abstract

Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this ‘semi-cardinal’ box-spline subspace satisfy certain boundary conditions extending the ‘not-a-knot’ end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4.

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References

  1. A. Bejancu, A new approach to semi-cardinal spline interpolation, East J. Approx. 6 (2000) 447–463.

    MATH  MathSciNet  Google Scholar 

  2. A. Bejancu, Semi-cardinal interpolation and difference equations: from cubic B-splines to a three-direction box-spline construction, Preprint, Department of Applied Mathematics, University of Leeds (2002).

  3. C.K. Chui, Multivariate Splines, CBMS-NSF Series in Applied Mathematics, Vol. 54 (SIAM, Philadelphia, PA, 1988).

    Google Scholar 

  4. C. de Boor, K. Höllig and S. Riemenschneider, Bivariate cardinal interpolation by splines on a three-direction mesh, Illinois J. Math. 29 (1985) 533–566.

    MATH  MathSciNet  Google Scholar 

  5. C. de Boor, K. Höllig and S. Riemenschneider, Box Splines (Springer, New York, 1993).

    MATH  Google Scholar 

  6. P.O. Frederickson, Quasi-interpolation, extrapolation and approximation on the plane, in: Proc. of the Manitoba Conf. on Numerical Mathematics, University of Manitoba, Winnipeg, 1971, pp. 159–167.

  7. L.S. Goldenstein, Tests for one-sided inverses of functions of several isometric operators and their applications, Soviet Math. Dokl. 5 (1964) 330–334.

    Google Scholar 

  8. L.S. Goldenstein and I.C. Gohberg, On a multidimensional integral equation on a half-space whose kernel is a function of the difference of the arguments, and on a discrete analogue of this equation, Soviet Math. Dokl. 1 (1960) 173–176.

    MathSciNet  Google Scholar 

  9. M.A. Sabin, The use of piecewise forms for the numerical representation of shape, Dissertation, Hungarian Academy of Sciences, Tanulmanyok 60/1977, MTA, Budapest (1977).

  10. M.A. Sabin and A. Bejancu, Boundary conditions for the 3-direction box-spline, in: Mathematics of Surfaces, Proc. of the 10th IMA Internat. Conf., Lecture Notes in Computer Science, Vol. 2768 (Springer, Berlin, 2003) pp. 244–261.

    Google Scholar 

  11. I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Part A: On the problem of smoothing of graduation, a first class of analytic approximation, Quart. Appl. Math. 4 (1946) 45–99.

    MathSciNet  Google Scholar 

  12. I.J. Schoenberg, Cardinal Spline Interpolation, CBMS – NSF Series in Applied Mathematics, Vol. 12 (SIAM, Philadelphia, PA, 1973).

    MATH  Google Scholar 

  13. I.J. Schoenberg, Cardinal interpolation and spline functions VI. Semi-cardinal interpolation and quadrature formulae, J. Anal. Math. 27 (1974) 159–204.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Leeds, LS2 9JT, UK

    Aurelian Bejancu

  2. Numerical Geometry Ltd., Lode, Cambridge, CB5 9EP, UK

    Malcolm A. Sabin

Authors
  1. Aurelian Bejancu
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  2. Malcolm A. Sabin
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Correspondence to Aurelian Bejancu.

Additional information

Communicated by C.A. Micchelli

AMS subject classification

41A15, 41A05, 41A25, 41A63, 39A70, 47B35

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Bejancu, A., Sabin, M.A. Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh. Adv Comput Math 22, 275–298 (2005). https://doi.org/10.1007/s10444-003-2601-7

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  • DOI: https://doi.org/10.1007/s10444-003-2601-7

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