# Harmonic Oscillator and Fourier Series

I am currently studying Fourier Series (on my own). I am using a few different references/sources. Some are more trying to give an intuition about Fourier Series and others are more rigorous.

Generally in references where explanation are not that rigorous, there is often an attempt at explaining the principle behind Fourier series, as making the signal resonate with an harmonic oscillator. Where if the signal "contains" that harmonic then the Fourier coefficient for this harmonic is different than 0 and 0 otherwise (I simplify).

My problem with this approach is that as I get deeper in the equations (again I am on my own so if I could ask a teacher probably he/she would answer this question), I really don't see the computation of the Fourier coefficient has something that has anything to do with the principle of an oscillator. The coefficients are an integral of the product of two functions. For me an oscillation is when you add up two waves together and get constructive or destructive interference.

I have seen the term "harmonic oscillator" being used in quite a few documents in which they were talking about Fourier series as well but:

1) I am not sure there's a direct connection. If there's one, could you please briefly tell me which one it is.

2) I don't think interpreting the principle by which a signal oscillates with a particular harmonic to explain the principle of Fourier series is a particularly accurate thing to say (at least it is misleading). I would like to know what experts think?

Thank you.

• The harmonics form what is called a orthonormal basis of $L^2$, so in order to extract their coefficients, it suffices to take the integral you describe. Have you tried looking at Fourier Analysis by Stein and Shakarchi? – Potato Oct 6 '13 at 23:36
• You are right that when you represent a function in terms of its Fourier series, you are representing it as sum of pure harmonics. Your understanding here is completely correct. – Potato Oct 6 '13 at 23:38
• @Potato. I don't know about this reference but I will find it. Thank you. Yes I am familiar with the idea that the cosine and sine of each term in the series are orthonormal. Interesting that you use the term "extract". My question is more about whether or not it is correct to say that the signal "resonates" for a given harmonics, if it contains that harmonic and if you can use this interpretation to give an intuition about how and why Fourier series works. – Marc Ourens Oct 6 '13 at 23:40
• Probably. It seems to be asking more about the physics of Fourier series, and whether the intuitive explanations are physically rigorous. However, it is a bit limiting to think of Fourier series only in terms of vibrations (or frequencies, sound, and related things). You can use Fourier series to solve the heat equation, for example, and many other things. – Potato Oct 6 '13 at 23:47
• What exactly do you mean by why? What exactly do you want explained? You might say it is just linear algebra: given any basis (for a Hilbert space...), you can represent an element as a sum of basis elements. The Fourier basis is special because it allows us to solve differential equations easily. – Potato Oct 6 '13 at 23:59