Discrete Fourier transform over a ring

In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. == Definition == Let R be any ring, let n ≥ 1 {\displaystyle n\geq 1} be an integer, and let α ∈ R {\displaystyle \alpha \in R} be a principal nth root of unity, defined by: The discrete Fourier transform maps an n-tuple ( v 0 , … , v n − 1 ) {\displaystyle (v_{0},\ldots ,v_{n-1})} of elements of R to another n-tuple ( f 0 , … , f n − 1 ) {\displaystyle (f_{0},\ldots ,f_{n-1})} of elements of R according to the following formula: By convention, the tuple ( v 0 , … , v n − 1 ) {\displaystyle (v_{0},\ldots ,v_{n-1})} is said to be in the time domain and the index j is called time.

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