Fast Fourier Transforms for Nonequispaced Data

  • Gerlind Plonka
  • Daniel Potts
  • Gabriele Steidl
  • Manfred Tasche
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.


  1. 9.
    A. Arnold, M. Bolten, H. Dachsel, F. Fahrenberger, F. Gähler, R. Halver, F. Heber, M. Hofmann, J. Iseringhausen, I. Kabadshow, O. Lenz, M. Pippig, ScaFaCoS - Scalable fast Coloumb solvers (2013).
  2. 14.
    R.F. Bass, K. Gröchenig, Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal. 36(3), 773–795 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 23.
    R.K. Beatson, W.A. Light, Fast evaluation of radial basis functions: methods for two–dimensional polyharmonic splines. IMA J. Numer. Anal. 17(3), 343–372 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 31.
    J.-P. Berrut, L.N. Trefethen, Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 32.
    G. Beylkin, On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal. 2(4), 363–381 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 33.
    G. Beylkin, R. Cramer, A multiresolution approach to regularization of singular operators and fast summation. SIAM J. Sci. Comput. 24(1), 81–117 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 35.
    Å. Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996)zbMATHCrossRefGoogle Scholar
  8. 48.
    M. Broadie, Y. Yamamoto, Application of the fast Gauss transform to option pricing. Manag. Sci. 49, 1071–1088 (2003)zbMATHCrossRefGoogle Scholar
  9. 61.
    E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris 346(9–10), 589–592 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 90.
    J. Duchon, Fonctions splines et vecteurs aleatoires. Technical report, Seminaire d’Analyse Numerique, Universite Scientifique et Medicale, Grenoble, 1975Google Scholar
  11. 93.
    A.J.W. Duijndam, M.A. Schonewille. Nonuniform fast Fourier transform. Geophysics 64, 539–551 (1999)CrossRefGoogle Scholar
  12. 95.
    A. Dutt, V. Rokhlin, Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Stat. Comput. 14(6), 1368–1393 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 96.
    A. Dutt, V. Rokhlin, Fast Fourier transforms for nonequispaced data II. Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 101.
    B. Elbel, Mehrdimensionale Fouriertransformation für nichtäquidistante Daten. Diplomarbeit, Technische Hochschule Darmstadt, 1998Google Scholar
  15. 102.
    B. Elbel, G. Steidl, Fast Fourier transform for nonequispaced data, in Approximation Theory IX (Vanderbilt University Press, Nashville, 1998), pp. 39–46zbMATHGoogle Scholar
  16. 103.
    A. Elgammal, R. Duraiswami, L.S. Davis, Efficient non-parametric adaptive color modeling using fast Gauss transform. Technical report, University of Maryland, 2001CrossRefGoogle Scholar
  17. 106.
    H.G. Feichtinger, K. Gröchenig, T. Strohmer, Efficient numerical methods in non-uniform sampling theory. Numer. Math. 69(4), 423–440 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 110.
    J.A. Fessler, B.P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 117.
    S. Foucart, A note guaranteed sparse recovery via 1-minimization. Appl. Comput. Harmon. Anal. 29(1), 97–103 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 118.
    S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2013)zbMATHCrossRefGoogle Scholar
  21. 120.
    K. Fourmont, Schnelle Fourier–Transformation bei nichtäquidistanten Gittern und tomographische Anwendungen. Dissertation, Universität Münster, 1999zbMATHGoogle Scholar
  22. 122.
    M. Frigo, S.G. Johnson, The design and implementation of FFTW3. Proc. IEEE 93, 216–231 (2005)CrossRefGoogle Scholar
  23. 123.
    M. Frigo, S.G. Johnson, FFTW, C subroutine library (2009).
  24. 147.
    L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, Cambridge, 1988)zbMATHGoogle Scholar
  25. 148.
    L. Greengard, P. Lin, Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9(1), 83–97 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 149.
    L. Greengard, J. Strain, The fast Gauss transform. SIAM J. Sci. Stat. Comput. 12(1), 79–94 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 150.
    L. Greengard, X. Sun, A new version of the fast Gauss transform, in Proceedings of the International Congress of Mathematicians (Berlin, 1998), Documenta Mathematica, vol. 3 (1998), pp. 575–584Google Scholar
  28. 176.
    J.I. Jackson, C.H. Meyer, D.G. Nishimura, A. Macovski, Selection of a convolution function for Fourier inversion using gridding. IEEE Trans. Med. Imag. 10, 473–478 (1991)CrossRefGoogle Scholar
  29. 177.
    M. Jacob, Optimized least-square nonuniform fast Fourier transform. IEEE Trans. Signal Process. 57(6), 2165–2177 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 198.
    J. Keiner, S. Kunis, D. Potts, Using NFFT3 - a software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Softw. 36, Article 19, 1–30 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 199.
    J. Keiner, S. Kunis, D. Potts, NFFT 3.4, C subroutine library. Contributor: F. Bartel, M. Fenn, T. Görner, M. Kircheis, T. Knopp, M. Quellmalz, T. Volkmer, A. Vollrath
  32. 211.
    S. Kunis, S. Kunis, The nonequispaced FFT on graphics processing units. Proc. Appl. Math. Mech. 12, 7–10 (2012)CrossRefGoogle Scholar
  33. 213.
    S. Kunis, D. Potts, Stability results for scattered data interpolation by trigonometric polynomials. SIAM J. Sci. Comput. 29(4), 1403–1419 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 214.
    S. Kunis, D. Potts, G. Steidl, Fast Gauss transforms with complex parameters using NFFTs. J. Numer. Math. 14(4), 295–303 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 224.
    J.-Y. Lee, L. Greengard, The type 3 nonuniform FFT and its applications. J. Comput. Phys. 206(1), 1–5 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 251.
    F. Nestler, Automated parameter tuning based on RMS errors for nonequispaced FFTs. Adv. Comput. Math. 42(4), 889–919 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 252.
    F. Nestler, Parameter tuning for the NFFT based fast Ewald summation. Front. Phys. 4(28), 1–28 (2016)Google Scholar
  38. 253.
    N. Nguyen, Q.H. Liu, The regular Fourier matrices and nonuniform fast Fourier transforms. SIAM J. Sci. Comput. 21(1), 283–293 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 256.
    A. Nieslony, G. Steidl, Approximate factorizations of Fourier matrices with nonequispaced knots. Linear Algebra Appl. 366, 337–351 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 270.
    M. Pippig, PNFFT, Parallel Nonequispaced FFT subroutine library (2011).
  41. 272.
    M. Pippig, D. Potts, Parallel three-dimensional nonequispaced fast Fourier transforms and their application to particle simulation. SIAM J. Sci. Comput. 35(4), C411–C437 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 279.
    D. Potts, Fast algorithms for discrete polynomial transforms on arbitrary grids. Linear Algebra Appl. 366, 353–370 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 280.
    D. Potts, Schnelle Fourier-Transformationen für nichtäquidistante Daten und Anwendungen. Habilitation, Universität zu Lübeck, 2003Google Scholar
  44. 281.
    D. Potts, G. Steidl, Fast summation at nonequispaced knots by NFFTs. SIAM J. Sci. Comput. 24(6), 2013–2037 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 291.
    D. Potts, G. Steidl, M. Tasche, Trigonometric preconditioners for block Toeplitz systems, in Multivariate Approximation and Splines (Mannheim, 1996) (Birkhäuser, Basel, 1997), pp. 219–234CrossRefGoogle Scholar
  46. 294.
    D. Potts, G. Steidl, M. Tasche, Fast Fourier transforms for nonequispaced data. A tutorial, in Modern Sampling Theory: Mathematics and Applications (Birkhäuser, Boston, 2001), pp. 247–270CrossRefGoogle Scholar
  47. 296.
    D. Potts, G. Steidl, A. Nieslony, Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98(2), 329–351 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 308.
    H. Rauhut, R. Ward, Sparse Legendre expansions via 1-minimization. J. Approx. Theory 164(5), 517–533 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 338.
    G. Steidl, A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3–4), 337–353 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 365.
    T. Volkmer, OpenMP parallelization in the NFFT software library. Preprint 2012-07, Faculty of Mathematics, Technische Universität Chemnitz (2012)Google Scholar
  51. 385.
    S.-C. Yang, H.-J. Qian, Z.-Y. Lu, A new theoretical derivation of NFFT and its implementation on GPU. Appl. Comput. Harmon. Anal. 44(2), 273–293 (2018)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gerlind Plonka
    • 1
  • Daniel Potts
    • 2
  • Gabriele Steidl
    • 3
  • Manfred Tasche
    • 4
  1. 1.University of GöttingenGöttingenGermany
  2. 2.Chemnitz University of TechnologyChemnitzGermany
  3. 3.TU KaiserslauternKaiserslauternGermany
  4. 4.University of RostockRostockGermany

Personalised recommendations