Parseval's identity

Statement

Let $f \in L^2([-\pi, \pi])$ be a square-integrable function with complex Fourier coefficients $c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx}\,dx$ for $n \in \mathbb{Z}$. Then the series of squared moduli of its Fourier coefficients converges and equals the normalized $L^2$ norm squared of $f$: $\sum_{n=-\infty}^{\infty} |c_n|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2\,dx$. More generally, for any orthonormal basis $(e_n)_{n \in I}$ of a Hilbert space $H$ and any $x \in H$, $\|x\|^2 = \sum_{n \in I} |\langle x, e_n \rangle|^2$.

Why is it true?

Parseval's identity is the infinite-dimensional Pythagorean theorem. Just as the squared length of a vector in $\mathbb{R}^3$ equals the sum of the squares of its projections onto three mutually perpendicular axes, the total 'energy' $\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2\,dx$ of a signal or wave equals the sum of the energies $|c_n|^2$ contained in each pure frequency $e^{inx}$. Because the harmonics $e^{inx}$ are mutually orthogonal, different frequencies do not interfere in the total energy, so transforming a function from the time domain to the frequency domain preserves its $L^2$ geometry exactly.

Topics that use this theorem

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Step-by-step proofs

No step-by-step proof yet for this theorem.