Let \(f(x)\) be defined for \(x \in[-L, L]\). Parseval's identity is given by

\[\frac{1}{L} \int_{-L}^{L} f^{2}(x) d x=\frac{a_{0}^{2}}{2}+\sum_{n=1}^{\infty} a_{n}^{2}+b_{n}^{2}\]

Assuming the the Fourier series of \(f(x)\) converges uniformly in \((-L, L)\), prove Parseval's identity by multiplying the Fourier series representation by \(f(x)\) and integrating from \(x=-L\) to \(x=L\). [We will encounter Parseval's equality for Fourier transforms, which is a continuous version of this identity.]