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What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)?


This question is based on the question of Kevin Lin, which didn't quite fit at Mathoverflow. Answers at any level of sophistication are welcome.

  • 12
    [Speaking of "for dummies", there's this book...](http://www.amazon.com/dp/354023165X)2011-05-08
  • 3
    When I was learning about FTs for actual work in signal processing, years ago, I found R. W. Hamming's book Digital Filters and Bracewell's The Fourier Transform and Its Applications good intros to the basics. Strang's Intro. to Applied Math. would be a good next step. Do a discrete finite FT by hand of a pure tone signal over a few periods to get a feel for the matched filtering and the relation of constructive and destructive interference to orthogonality.2012-05-09
  • 1
    Here's a neat [post](http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/) breaking down the Fourrier transform.2014-03-23
  • 0
    I used [this book](http://www.amazon.com/Who-Fourier-Mathematical-Transnational-College/dp/0964350408/ref=sr_1_3?s=books&ie=UTF8&qid=1395780570&sr=1-3&keywords=college+of+lex) to both learn calculus and the Fourier transform in high school. It was outstanding.2014-03-25
  • 3
    Here's a video I made a while ago describing the fourier series and fourier transform. It's a bit of a light hearted approach. http://www.youtube.com/watch?v=Qm84XIoTy0s2014-03-26
  • 3
    Let me leave a link to this awesome Fourier transformation explanation for Russian users here: http://habrahabr.ru/company/achiever/blog/204956/2014-04-29
  • 0
    There is a related question on DSP.SE: [Why is the Fourier transform so important?](http://dsp.stackexchange.com/q/69/9101)2014-07-02
  • 0
    Your question where to read. And my quesiton - where I can write about it. http://math.stackexchange.com/questions/1545143/where-can-i-write-and-publish-an-article-about-fourier-transform2015-11-25
  • 0
    I have wrote an article "FT Faq" here: https://sites.google.com/site/burlachenkok/articles/faq_ft It is in Russian, so please use google translate.2016-01-23
  • 0
    Related: https://math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/2016-12-04

14 Answers 14