Applications
This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized quickly using the fast Fourier transform, and this yields a fast method for solving systems of linear equations with circulant matrices. Similarly, the Fourier transform on arbitrary groups can be used to give fast algorithms for matrices with other symmetries (Åhlander & Munthe-Kaas 2005). These algorithms can be used for the construction of numerical methods for solving partial differential equations that preserve the symmetries of the equations (Munthe-Kaas 2006).
Read more about this topic: Fourier Transform On Finite Groups