What is a good reference for introducing non-commutative fourier transform for Electrical Engineers and Theoretical Computer Scientists in an explicit way?
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2$\begingroup$ What do you mean by non-com. FT ? Expansion by group characters or matrix elements ? What is the use of this in EE ? $\endgroup$– Alexander ChervovCommented Jun 5, 2012 at 5:57
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1$\begingroup$ I join Alexander's petition for clarification. Unless representation theory is studied in EE, the question is not very clear to me. $\endgroup$– Juan Bermejo VegaCommented Jun 5, 2012 at 9:39
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$\begingroup$ @Juan Representation Theory is useful in coding theory (space time codes) $\endgroup$– TurboCommented Jun 5, 2012 at 21:52
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$\begingroup$ I think representation theory of finite groups is exactly what's needed. For some strange reason the actual Fourier transform formulas are never given (e.g. in Serre's book) $\endgroup$– Igor RivinCommented Jun 5, 2012 at 23:09
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1$\begingroup$ @Alexander I am unsure of that. I just am not trained in their language. $\endgroup$– TurboCommented Jun 7, 2012 at 20:49
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6 Answers
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I think that "Group representations in probability and statistics" by Persi Diaconis is a very good choice. It is really intended for non-algebraists and is well motivated by real world examples. And it is available for free at project Euclid.
You could also try my book which is intended for a 4th year math undergrad or first year grad course and addresses some applications (sorry to plug my own book). It is very explicit (you won't see the word module).
answered Jun 5, 2012 at 0:42
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I think the book Engineering Applications of Noncommutative Harmonic Analysis by Chirikjian and Kyatkin might be exactly what you are looking for. Although I haven't read very much of it, the sections that I have read are very nice and seem mathematically rigorous, yet geared toward applications.
Also, there are a couple of excellent books by Myoung An and Richard Tolimieri about harmonic analysis over finite groups. The first is called Time-Frequency Representations, and it's about harmonic analysis over finite abelian groups with applications to audio signal processing. The second is called Group Filters and Image Processing, and it's about harmonic analysis over finite nonabelian groups with applications to image processing. I can't say enough about these books. The mathematical presentation is rigorous and elegant, and the applied examples are very explicit, including Matlab code and demonstrating how the authors have applied the techniques in work they have done for defense contractors. An overview of some of the material presented in these two book appears in this paper.
I highly recommend the books by Tolimieri and An -- especially "Time-Frequency Representations" -- to anyone who wants to learn signal processing the "right" way. (Incidentally, these authors have a new book called Ideal Sequence Design in Time-Frequency Space, but I have not read it.)
answered Jun 5, 2012 at 2:12
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$\begingroup$ Looks like Myong An is fromPrometheus:) $\endgroup$– TurboCommented Jun 13, 2012 at 22:04
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$\begingroup$ Actually, Prometheus is a company that Myoung and Richard started in order to publish their books and make them available at much more reasonable prices than the big publishers were willing to accept. $\endgroup$ Commented Jun 22, 2012 at 5:02
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Barry Simon's representation theory of finite and compact groups is good.
answered Jun 4, 2012 at 21:29
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1$\begingroup$ The title is actually "Representations of Finite and Compact Groups" $\endgroup$– StoppleCommented Jun 4, 2012 at 21:35
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$\begingroup$ Hi Igor: Is the book explicit enough for a computer scientist or an EE to understand and work through to get maturity? $\endgroup$– TurboCommented Jun 4, 2012 at 23:49
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1$\begingroup$ I think it is pretty concrete, though I have not had any guinea pigs, er, electrical engineers, to test it on. $\endgroup$ Commented Jun 5, 2012 at 0:19
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$\begingroup$ @Igor Is there a notion of Fourier transform in incidence algebras? $\endgroup$– TurboCommented Jun 6, 2012 at 0:23
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Perhaps you want to have a look at Lang "$SL_2(\mathbb{R})$" - Chapter V Spherical transform and Chapter VIII Plancherel formula: His approach is a classical, global one, which is probably more digestive than an infinitesimal approach at first encounter. Global approach means that you work only on the group (and not on its Lie algebra), and prove everything via integral transforms, Mellin transforms and special functions, so only a little bit of advanced calculus is needed.
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$\begingroup$ Would this help in non-commutative setting? $\endgroup$– TurboCommented Jun 13, 2012 at 22:02
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2$\begingroup$ What is a non-commutative setting and by what should it help? SL(2, R) is non-commutative, and the Harish-Chandra transform (for reductive groups only) and the Plancherel theorems are the only generalization of abelian Fourier analysis, I know. $\endgroup$ Commented Jun 15, 2012 at 10:19
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I am responding to the question, "What is a good reference for introducing non-commutative fourier transform for Electrical Engineers"
As an introduction to the subject, electrical engineers might find it helpful to look at the book, "Symmetries and Groups in Signal Processing", authored by me, and published by Springer.
Virendra P. Sinha
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I think $\S$ 16 of the book "Elements of the Theory of Representations" by A.A. Kirillov might be relevant. In $\S$ 16 he discusses concrete examples of representations of finite groups. Thus in $\S$ 16.1 he studies basic harmonic analysis on 3-dimensional cube, and even obtains some quantitative estimates using the representation theory of the group of symmetries of the cube.
