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

Authors: Kwong-Ip Liu, Josef Dick and Fred J. Hickernell
Journal: Math. Comp. 78 (2009), 1573-1591
MSC (2000): Primary 42C10, 41A15
Published electronically: January 9, 2009
MathSciNet review: 2501064
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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 $\mathcal {O}(N \log N)$ 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.

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Additional Information

Kwong-Ip Liu
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China

Josef Dick
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

Fred J. Hickernell
Affiliation: Department of Applied Mathematics, Illinois Institute of Technology, Room E1-208, 10 W. 32nd Street, Chicago, Illinois 60616
ORCID: 0000-0001-6677-1324

Received by editor(s): January 11, 2008
Received by editor(s) in revised form: July 29, 2008
Published electronically: January 9, 2009
Additional Notes: This research was supported in part by the Hong Kong Research Grants Council grant HKBU/2009/04P, the HKSAR Research Grants Council Project No. HKBU200605 and the United States National Science Foundation grant NSF-DMS-0713848.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.