The author defines the generalized discrete Fourier transform (GDFT (\({\mathbb{F}}G))\) as being the mapping \(\sigma_ G: {\mathbb{F}}G\to A_ 1\oplus...\oplus A_ s\) which decomposes the semisimple group algebra \({\mathbb{F}}G\) into simple Wedderburn components \(A_ i\), \(i=1,...,s\). The GDFT also satisfies the known properties of the DFT (inversion, convolution, phase shift, Parseval-Plancherel-identity).
The associated mappings \(\sigma_ i: {\mathbb{F}}G\to A_ i\) are the irreducible representations of the group algebra which plays the role of a universal \({\mathbb{F}}G\)-module in which each other \({\mathbb{F}}G\)-module occurs. Since in many applications such as coding theory, signal processing and picture processing, the set of data forms an \({\mathbb{F}}G\)- module, the importance of the study of irreducible representations is clear from the point of view of applications. The computational aspects are analyzed and fast versions of the GDFT are described in case \({\mathbb{F}}\) is a splitting field or a prime field.