Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform

@article{Morgenstern1973NoteOA,
  title={Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform},
  author={Jacques Morgenstern},
  journal={J. ACM},
  year={1973},
  volume={20},
  pages={305-306}
}
A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound <italic>c</italic> can be assigned to the modulus of the complex numbers involved in the computation. In the case of the fast Fourier transform, the lower bound is (<italic>n</italic>/2) log<subscrpt>2</subscrpt><italic>n</italic> when <italic>c</italic> = 1. 
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References

SHOWING 1-5 OF 5 REFERENCES
ON LINEAR ALGORITHMS
An algorithm for the machine calculation of complex Fourier series
TLDR
Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
An algorithm for the machine computation of complex
  • Fourier series. Math. Comp
  • 1964
Algorithmes lindaires
  • Compt. Rend. Acad. Sci