A multivariate fast discrete walsh transform with an application to function interpolation

Kwong Ip LIU*, Josef Dick, Fred J. Hickernell

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

2 Citations (Scopus)


For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only O{script}(N logN) operations, where N is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.

Original languageEnglish
Pages (from-to)1573-1591
Number of pages19
JournalMathematics of Computation
Issue number267
Publication statusPublished - Jul 2009

Scopus Subject Areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A multivariate fast discrete walsh transform with an application to function interpolation'. Together they form a unique fingerprint.

Cite this