My question has two parts:

$1)$ Could anyone explain in simple terms what a Fourier Transform is?

$2)$ What are some of the applications of Fourier Analysis in the field of high school mathematics?

  • $\begingroup$ Re 2), the Fourier transform is a mathematical way to express the spectrum of a signal such as an audio signal -- think of the use of an equalizer for example. If high school physics includes spectra, then ideally high school calculus would complement it by giving the relevant math, but that seems a bit difficult in practice. $\endgroup$ – ForgotALot Apr 2 '16 at 7:09
  • $\begingroup$ Related: math.stackexchange.com/questions/579453/… $\endgroup$ – Watson Dec 4 '16 at 12:59

So let's say that we have a function which represents something interesting (physically or mathematically). The function could be simple or more likely very complicated. The Fourier transform changes the way we look at the function. Instead of in the complicated form we look at it as a sum of sines and cosines. The transform is then said to represent the spectrum of of the original function. That may not sound important but the spectrum tells us what frequencies are very relevant and which are less so.

An example that I always think of are those electronic keyboards. A piano generates sound by striking a string and (if properly tuned) have a series of harmonic waves which fit the boundary conditions of the string which make a particular tone. In order to replicate that, we cannot electronically play the infinite set of harmonics found in nature, instead we find the most dominate frequencies replicate those specifically and ignore the rest. The frequency spectrum, and therefore the Fourier transform, is a way of telling us what is important to a function (at least when trying to think of it as sines and cosines).

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