This is my first post so please go easy, I don't know all the rules yet.
I was reviewing Fourier series for the 5th time and I realized that every explanation I read goes into the orthogonality of Sin and Cosine with some integrals and jumps into the formulae for determining the coefficients.
I don't understand how those formulae work to magically produce the coefficients and how this is related to the orthogonality of Sin and Cosine. I'm giving Fourier series a last shot for intuitive understanding before I accept it as fact and rote learn the formulae, please help!
Thank you
Best Answer
It's enlightening to think about an analogous idea in linear algebra. Suppose $\{v_1,v_2,\ldots,v_N \}$ is an orthonormal set of vectors in $\mathbb R^N$. Suppose also that $x \in \mathbb R^N$, and we want to write $x$ as a linear combination of the vectors $v_1,\ldots, v_N$: \begin{equation} x = \sum_{i=1}^N c_i v_i. \end{equation} How can we determine the coefficients in this linear combination? A trick that lets you determine the coefficients very easily is to take the inner product of both sides with $v_j$: \begin{align} \langle x, v_j \rangle &= \left \langle \sum_{i=1}^N c_i v_i, v_j \right \rangle \\ &= \sum_{i=1}^N c_i \langle v_i, v_j \rangle \\ &= c_j. \end{align} (Notice that in the last step, all but one of the terms vanishes, which is awesome.)
So we have discovered that \begin{equation*} c_j = \langle x, v_j \rangle. \end{equation*}
This great trick is called the "Fourier trick". (At least, that's what David Griffiths calls it in his physics books.) The same trick allows you to easily compute the coefficients in a Fourier series.