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mkoyama-2007-prop

Course: MATH 197, Fall 2009
School: Harvey Mudd College
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Word Count: 415

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Thesis Proposal: Discrete Fourier Transforms for Finite Groups Masanori Koyama April 5, 2006 In signal processing, one may improve the e ciency of data analysis, in particular of convolution, by using a Fourier basis via a discrete Fourier transform (DFT). Data points in the signal live in the C-algebra CZn , and the multiplication of elements in CZn can be considered a module action of CZn on CZn . This action...

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Thesis Proposal: Discrete Fourier Transforms for Finite Groups Masanori Koyama April 5, 2006 In signal processing, one may improve the e ciency of data analysis, in particular of convolution, by using a Fourier basis via a discrete Fourier transform (DFT). Data points in the signal live in the C-algebra CZn , and the multiplication of elements in CZn can be considered a module action of CZn on CZn . This action can then be interpreted as linear transformation in n dimensional vector space over C, which is expressible as matrix. With respect to the Fourier basis, this matrix is diagonal. Representation theory and algebraic complexity theory allowed for the extension of this idea to data points living in the group algebra CG for general group G. The basis of this generalization is Wedderburn s Theorem which states that CG is isomorphic to a particular block diagonal matrix over C. The isomorphism claimed in this statement is the generalized DFT, which is the change of basis from a standard basis to the Fourier basis. Any e cient application of DFT is called <a href="/keyword/fast-fourier-transform/" ><a href="/keyword/fast-fourier/" >fast fourier</a> transform</a> . The cases for abelian groups are well studied; however, there are still many open questions for the case of non-abelian groups. The most prevalent constructions of DFTs are decimation in time algorithms. However, in this research I would deploy the decimation in frequency algorithm pioneered by Michael Orrison, which has been proving to be more e cient by the predecessor students of this research. I will attempt to devise an e cient way of implementing a DFT on speci c nite groups that are yet to be investigated, or to generalize the results by previous researchers on the studied nite groups. I have been exposed to the Fourier transform in a core engineering class (Enr59), Applied Analysis (Math 180), and Mathematical Analysis II. The algebraic approach to the project is going to be fueled by coursework in Algebra II (Math 172), and Applied Representation Theory (Math 196). Representation theory in particular is the core of the eld and hence Math 196 will provide a rm basis for investigating the literature. As for the complexity aspect of the project, I have researched the linear complexity of graphs, which provided some familiarity to the eld. I will have extensively read <a href=&...

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