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Error Analysis of Splines for Periodic Problems Using Lattice Designs

In: Monte Carlo and Quasi-Monte Carlo Methods 2004

Author

Listed:
  • Xiaoyan Zeng

    (Hong Kong Baptist University, Department of Mathematics)

  • King-Tai Leung

    (Hong Kong Baptist University, Department of Mathematics)

  • Fred J. Hickernell

    (Hong Kong Baptist University, Department of Mathematics)

Abstract

Summary Splines are minimum-norm approximations to functions that interpolate the given data, (xi, f(xi)). Examples of multidimensional splines are those based on radial basis functions. This paper considers splines of a similar form but where the kernels are not necessarily radially symmetric. The designs, {xi}, considered here are node-sets of integration lattices. Choosing the kernels to be periodic facilitates the computation of the spline and the error analysis. The error of the spline is shown to depend on the smoothness of the function being approximated and the quality of the lattice design. The quality measure for the lattice design is similar, but not equivalent to, traditional quality measures for lattice integration rules.

Suggested Citation

  • Xiaoyan Zeng & King-Tai Leung & Fred J. Hickernell, 2006. "Error Analysis of Splines for Periodic Problems Using Lattice Designs," Springer Books, in: Harald Niederreiter & Denis Talay (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2004, pages 501-514, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-31186-7_31
    DOI: 10.1007/3-540-31186-6_31
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