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I have a simple question:

What is the (analytical) intuition behind the Fourier series:

$f(t)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}\left(a_{k}\cos\left(kt\right)+b_{k}\sin\left(kt\right)\right)$

I have read in an article that this expression results from $f(t)=\sum_{k=1}^{\infty} d_n\cos (nt+\phi_n)$ using some trigonometric theorems. But the question is still: why can I write every function as the second expression?

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    $\begingroup$ I think you should just keep going, intuition will come. This is NOT obvious at all, otherwise it would not have been such a great breakthrough. $\endgroup$ – Giuseppe Negro Nov 17 '17 at 9:35
  • $\begingroup$ You can read this: betterexplained.com/articles/… $\endgroup$ – Dove Nov 17 '17 at 9:58
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    $\begingroup$ Arguably, there is no good analytical intuition for being able to write $f$ as a Fourier series. When Fourier first claimed such a thing, he was banned from publishing for over a decade because the prominent Mathematicians at the time thought his claim was blatantly false. $\endgroup$ – Disintegrating By Parts Nov 17 '17 at 16:14
  • $\begingroup$ Actually you can't "write every function" in general. For example, if $f(t)$ is not periodic it is impossible. Also, convergence can be tricky if $f(t)$ is not integrable. See also the Gibbs phenomenon. $\endgroup$ – Somos Nov 18 '17 at 0:19
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I assume that you are familiar with Taylor expansion. In principle this just says that all (analytic) functions can be written as a sum of polynomials. This is in the beginning not trivial at all, but seems to work just fine. I assume that you got used to this fact.

Same thing can be said about the Fourierseries. It's not obvious that this is indeed possible, but it might just work (and as mathematics show, it does indeed work). Fourierseries is in a way similar to Taylor series. We write a function as an infinite sum of other functions.

The space of all functions can be seen as a vector space (if you don't know this, maybe just ignore what's coming next). We can try to find, as with any vector space, a base. Apparently all the sin and cosine fucntions happen to be a base for this vector space.

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Any function $ f(x)$, between limits $ a $ and $b$ can be expressed as sum of sinusoidals . This is based on the same logic as that of the Taylor/ Maclaurin series, but here for a bounded range of input.

You can basically prove it in reverse by determining the values of the constants by differentiating on both sides w.r.t x and substituting x=0.

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The reason it works is that ∫ fn*(f1+f2+f3+...) = ∫ (fnf1) + ∫ (fnf2) + ∫ (fn*f3) +....and multiplying any 2 sine waves of frequency n1 and frequency n2 (of period p/n1 and p/n2) and integrating over one period p will always equal zero unless n1=n2. See graph of sin2(x)

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