The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno. Gauss wanted to interpolate the … See more

Let $${\displaystyle x_{0},\ldots ,x_{n-1}}$$ be complex numbers. The DFT is defined by the formula
where See more

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry
$${\displaystyle X_{n-k}=X_{k}^{*}}$$
and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). … See more

As defined in the multidimensional DFT article, the multidimensional DFT
transforms an array … See more

Cooley–Tukey algorithm
By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT of any composite size $${\textstyle n=n_{1}n_{2}}$$ into many smaller DFTs of sizes See more

Bounds on complexity and operation counts
A fundamental question of longstanding theoretical interest is … See more

An $${\textstyle O(n^{5/2}\log n)}$$ generalization to spherical harmonics on the sphere S with n nodes was described by Mohlenkamp, along with an algorithm conjectured (but … See more