# References¶

Please cite the following two papers if you use this software:

[FIN] A parallel non-uniform fast Fourier transform library based on an “exponential of semicircle” kernel. A. H. Barnett, J. F. Magland, and L. af Klinteberg. SIAM J. Sci. Comput. 41(5), C479-C504 (2019). arxiv version

[B20] Aliasing error of the exp\((\beta \sqrt{1-z^2})\) kernel in the nonuniform fast Fourier transform. A. H. Barnett. Appl. Comput. Harmon. Anal. 51, 1-16 (2021). arxiv version

## Background references¶

For the Kaiser–Bessel kernel and the related PSWF, see:

[KK] Chapter 7. System Analysis By Digital Computer. F. Kuo and J. F. Kaiser. Wiley (1967).

[FT] K. Fourmont. Schnelle Fourier-Transformation bei nichtäquidistanten Gittern und tomographische Anwendungen. PhD thesis, Univ. Münster, 1999.

[F] Non-equispaced fast Fourier transforms with applications to tomography. K. Fourmont. J. Fourier Anal. Appl. 9(5) 431-450 (2003).

[FS] Nonuniform fast Fourier transforms using min-max interpolation. J. A. Fessler and B. P. Sutton. IEEE Trans. Sig. Proc., 51(2):560-74, (Feb. 2003)

[ORZ] Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation. A. Osipov, V. Rokhlin, and H. Xiao. Springer (2013).

[KKP] Using NFFT3—a software library for various nonequispaced fast Fourier transforms. J. Keiner, S. Kunis and D. Potts. Trans. Math. Software 36(4) (2009).

[DFT] How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix? A. H. Barnett, submitted, SIAM Rev. (2020). arxiv version

The appendix of the last of the above contains the first known published proof of the Kaiser–Bessel Fourier transform pair. This next two papers prove error estimates for sinh-type and other kernels closely related (and possibly slightly more optimal) than ours:

[PT] Uniform error estimates for the NFFT. D. Potts and M. Tasche. (2020). arxiv

[PT2] Continuous window functions for NFFT. D. Potts and M. Tasche. (2020). arxiv. In revision, Adv. Comput. Math.

In late 2020 it was pointed out to us by Piero Angeletti that the exponential of semicircle kernel developed for FINUFFT had in fact been independently proposed:

[AN] A new window based on exponential function. K. Avci and A. Nacaroğlu. 2008 Ph.D. Research in Microelectronics and Electronics, Istanbul. 69-72 (2008). doi:10.1109/RME.2008.4595727.

FINUFFT builds upon the CMCL NUFFT, and the Fortran wrappers are very similar to its interfaces. For that, the following are references:

[GL] Accelerating the Nonuniform Fast Fourier Transform. L. Greengard and J.-Y. Lee. SIAM Review 46, 443 (2004).

[LG] The type 3 nonuniform FFT and its applications. J.-Y. Lee and L. Greengard. J. Comput. Phys. 206, 1 (2005).

Inversion of the NUFFT is covered in [KKP] above, and in:

[GLI] The fast sinc transform and image reconstruction from nonuniform samples in \(\mathbf{k}\)-space. L. Greengard, J.-Y. Lee and S. Inati, Commun. Appl. Math. Comput. Sci (CAMCOS) 1(1) 121-131 (2006).

The original NUFFT analysis using truncated Gaussians is (the second improving upon the convergence rate of the first):

[DR] Fast Fourier Transforms for Nonequispaced data. A. Dutt and V. Rokhlin. SIAM J. Sci. Comput. 14, 1368 (1993).

[S] A note on fast Fourier transforms for nonequispaced grids. G. Steidl, Adv. Comput. Math. 9, 337-352 (1998).

## Talk slides¶

These PDF slides may be a useful introduction.