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--- title: "Fourier formula" source: https://www.jemoka.com/posts/kbhlearn_more/ --- For vector \(v\) in the span of orthogonal basis \(v_1, ..v_{n}\): \begin{equation} v = c_1 v_1 + \dots + c_{n} v_{n} \end{equation} we can write: \begin{equation} c_{j} = \frac{v \cdot v_{j}}{ v_{j} \cdot v_{j}} \end{equation} Proof: \begin{equation} \langle v, v_{j} \rangle = c_{n} \langle v_{1}, v_{j} \rangle \dots \end{equation} which is \(0\) for all cases that’s not \(\langle v_{j}, v_{j} \rangle\) as the \(v\) are orthogonal, and \(\mid v_{j} \mid^{2}\) for the case where it is. Hence, we see that: \begin{equation} \langle v, v_{j} \rangle = c_{j} \mid v_{j}\mid^{2} \end{equation} Which gives: \begin{equation} c_{j} = \frac{\langle v,v_{j} \rangle}{\mid v_{j}\mid^{2}} \end{equation} as desired.