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It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat equations, and its very evident that present world's seminal mathematician J.Tate has also used the Fourier analysis in his celebrated thesis.

So I was looking for some most elegant explanation about the beauty of Fourier analysis giving an intuition into the subject, and for the sake of brevity, assume that the targeted naive reader has some basic mathematical knowledge, ( not so naive !! ). I think the most difficult job is to explain things naively. But there are many people who can explain it mathematically taking a function and applying it and showing the effect, but its the most tough job to do it other way, as remarked by Prof.Einstein, " If you can't explain simply, you don't understand it well-enough " .

I would be very happy if someone maps the theory to the nature and explain the theory in terms of things going on around us, like trees, leaves etc..( not necessarily trees and leaves, but in any way ) and how could one apply them to the nature practically.

Thanks a lot !!

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I can imagine some nice 3D graphics. – Michael Hardy Mar 13 '12 at 11:49
@MichaelHardy : But do you have any links that will show us what you are thinking about ? – Iyengar Mar 20 '12 at 2:43
No. Do you know any graphics experts who might want to collaborate on something like this? – Michael Hardy Mar 20 '12 at 18:46
Maybe this is what you are looking for. – Michael Greinecker Sep 5 '12 at 22:39
@Iyengar I haven't read the book myself, so I don't feel comfortable endorsing it in an answer. – Michael Greinecker Sep 6 '12 at 9:11
up vote 7 down vote accepted

See Princeton Companion to Mathematics (I'm including some more tangential topics since you tagged the question the way you did):

  • Section III.27 Fourier Transform
  • Section III.36 The Heat Equation
  • Section III.89 Spherical Harmonics
  • Section III.93 Transforms
  • Section III.100 Wavelets
  • Section IV.2 Analytic Number Theory
  • Section IV.9 Representation Theory (especially subsection 3)
  • Section IV.11 Harmonic analysis
  • Section VI.25 Jean-Baptiste Joseph Fourier
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But for the sake of curiosity sir, does it explain it in a non-technical format ? . I can get technical description from text-books and other things, but one day a friend of mine asked me to describe it naively. So I wanted to do the same sir, Anyway thanks a lot sir. +1 for elegant reference. – Iyengar Mar 13 '12 at 16:02

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