Generalizing the discrete Fourier transform
type: Item
instance of: scholarly article
title: Generalizing the discrete Fourier transform
zbMATH Open document ID: 0618.65145
DOI: 10.1016/0012-365X(85)90016-0
author: Thomas Beth
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 94B35
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 20C35
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 4003480
zbMATH Keywords: generalized discrete Fourier transform
zbMATH Keywords: semisimple group algebra
zbMATH Keywords: simple Wedderburn components
zbMATH Keywords: inversion
zbMATH Keywords: convolution
zbMATH Keywords: phase shift
zbMATH Keywords: Parseval- Plancherel-identity
zbMATH Keywords: irreducible representations
zbMATH Keywords: coding theory
zbMATH Keywords: signal processing
zbMATH Keywords: picture processing
zbMATH Keywords: splitting field
zbMATH Keywords: prime field
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4003480