# 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.
116 Citations
The arithmetic computational complexity of linear transforms
Quadratic and superquadratic estimates are obtained for the computational complexity of some linear transforms by circuits over the base {x + y} ∪ {ax: |a| ≤ C} consisting of addition and scalar
A Lower Bound for Fourier Transform Computation in a Linear Model Over 2x2 Unitary Gates Using Matrix Entropy
• Nir Ailon
• Computer Science
Chic. J. Theor. Comput. Sci.
• 2013
The main argument concluded from this work is that a potential function that might eventually help proving the \$\Omega(n\log n)\$ conjectured lower bound for computation of Fourier transform is not related to matrix determinant, but rather to a notion of matrix entropy.
Lower bounds on the bounded coefficient complexity of bilinear maps
• Mathematics, Computer Science
JACM
• 2004
Lower bounds of order n log n are proved for both the problem of multiplying polynomials of degree n, and of dividing polynomers with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers, to establish a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.
Optimal Complexity Lower Bound for Polynomial Multiplication
• Mathematics, Computer Science
• 2003
Lower bounds of order n log n are proved for both the problem of multiplying polynomials of degree n, and of dividing polynomers with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers, to establish a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.
An n\log n Lower Bound for Fourier Transform Computation in the Well Conditioned Model
The main technical contribution is an extension of matrix entropy used in Ailon (2013) for unitary matrices to a potential function computable for any matrix, using Shannon entropy on "quasi-probabilities".
A SPECTRAL APPROACH TO LOWER BOUNDS WITH APPLICATIONS TO GEOMETRIC SEARCHING
It is shown that summing up the weights of n (weighted) points within n halfplanes requires Ω(n logn) additions and subtractions, the first nontrivial lower bound for range searching over a group.
An Omega((n log n)/R) Lower Bound for Fourier Transform Computation in the R-Well Conditioned Model
This work eliminates the scaling restriction and provides a lower bound for computing any scaling of the Fourier transform, and allows the computational model to use extra memory.
A spectral approach to lower bounds
• B. Chazelle
• Mathematics, Computer Science
Proceedings 35th Annual Symposium on Foundations of Computer Science
• 1994
This work develops a general, entropy-based, method for relating the linear circuit complexity of a linear map A to the spectrum of A/sup T/A and shows that summing up the weights of n points within n halfplanes requires /spl Omega/(n log n) additions and subtractions.

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