Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path alg...

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Bibliographic Details
Published inThe Journal of fourier analysis and applications Vol. 24; no. 1; pp. 226 - 284
Main Authors Maslen, David, Rockmore, Daniel N., Wolff, Sarah
Format Journal Article
LanguageEnglish
Published New York Springer US 01.02.2018
Springer
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Algebra
Algorithms
Approximations and Expansions
Computation
Configurations
Fast Fourier transformations
Fourier Analysis
Fourier transforms
Group theory
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
Upper bounds
65T50
Generalized Fourier transform
Path algebra
43A30
Bratteli diagram
Computational complexity
20C15
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Summary:We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel’fand–Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-016-9516-4