Abstract
The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in \({\cal O}(N^d \log N)\) arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only \({\cal O}(N \log^d N)\) arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.
Similar content being viewed by others
References
G. Baszenski and F.-J. Delvos, A discrete Fourier transform scheme for Boolean sums of trigonometric operators, in: Multivariate Approximation Theory IV, ISNM 90, eds. C.K. Chui, W. Schempp and K. Zeller (Birkhäuser, Basel, 1989) pp. 15–24.
G. Beylkin, On the fast Fourier transform of functions with singularities, Appl. Comput. Harmon. Anal. 2 (1995) 363–381.
H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numerica 13 (2004) 147–269.
J.W. Cooley and J.W. Tukey, An algorithm for machine calculation of complex Fourier series, Math. Comput. 19 (1965) 297–301.
A.J.W. Duijndam and M.A. Schonewille, Nonuniform fast Fourier transform, Geophysics 64 (1999) 539–551.
A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data, SIAM J. Sci. Stat. Comput. 14 (1993) 1368–1393.
B. Elbel and G. Steidl, Fast Fourier transform for nonequispaced data, in: Approximation Theory IX, eds. C.K. Chui and L.L. Schumaker (Vanderbilt University Press, Nashville, 1998).
M. Fenn and D. Potts, Fast summation based on fast trigonometric transforms at nonequispaced nodes, Numer. Linear Algebra Appl. 12 (2005) 161–169.
J. Fessler and B. Sutton, NUFFT – nonuniform FFT toolbox for Matlab. http://www.eecs.umich.edu/fessler/code/index.html, 2002.
M. Frigo and S.G. Johnson, FFTW, a C subroutine library. http://www.fftw.org/.
K. Hallatschek, Fourier transformation auf dünnen Gittern mit hierarchischen Basen, Numer. Math. 63 (1992) 83–97.
S. Kunis and D. Potts, NFFT, Softwarepackage, C subroutine library. http://www.math.uni-luebeck.de/potts/nfft, 2002–2005.
D. Potts, G. Steidl and M. Tasche, Fast Fourier transforms for nonequispaced data: A tutorial, in: Modern Sampling Theory: Mathematics and Applications, eds. J.J. Benedetto and P.J.S.G. Ferreira (Birkhäuser, Boston, 2001) pp. 247–270.
F. Sprengel, A class of function spaces and interpolation on sparse grids, Numer. Funct. Anal. Optim. 21 (2000) 273–293.
G. Steidl, A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9 (1998) 337–353.
A.F. Ware, Fast approximate Fourier transforms for irregularly spaced data, SIAM Rev. 40 (1998) 838–856.
C. Zenger, Sparse grids, in: Parallel algorithms for partial differential equations (Kiel, 1990), volume 31 of Notes Numer. Fluid Mech. (Vieweg, Braunschweig, 1991) pp. 241–251.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Brezinski
Rights and permissions
About this article
Cite this article
Fenn, M., Kunis, S. & Potts, D. Fast evaluation of trigonometric polynomials from hyperbolic crosses. Numer Algor 41, 339–352 (2006). https://doi.org/10.1007/s11075-006-9017-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-006-9017-7
Key words and phrases
- trigonometric approximation
- hyperbolic cross
- sparse grids
- fast Fourier transform for nonequispaced knots
- NFFT
- FFT