Lagmental Vicfred

Parseval Turns L2 Energy into Fourier Coefficients by Vicfred

tags: mathematics, Fourier analysis, convolution, Sobolev spaces

The squared L2 norm of a periodic function equals the sum of squared Fourier coefficients. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Notation

Fourier analysis expands a function into frequencies. On the circle the coefficients are \(\widehat f(n)\); on \(\mathbf R\) the transform \(\widehat f(\xi)\) is an integral against an oscillatory exponential.

$$ \widehat f(n)=\frac1{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \frac1{2\pi}\int_{-\pi}^{\pi}|f(x)|^2\,dx=\sum_{n\in\mathbf Z}|\widehat f(n)|^2 $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \left\|f-\sum_{|n|\le N}\widehat f(n)e^{inx}\right\|_{L^2}^2=\sum_{|n|>N}|\widehat f(n)|^2\to0 $$

Interpretation

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad \widehat f(n)=\frac1{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx,\\[5pt] \mathsf{C}\;&:\quad \frac1{2\pi}\int_{-\pi}^{\pi}|f(x)|^2\,dx=\sum_{n\in\mathbf Z}|\widehat f(n)|^2. \end{aligned} $$

Limit of the argument

Normalization conventions move factors of \(2\pi\) between the transform, inverse transform, derivative rule, and Gaussian formula.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \frac1{2\pi}\int_{-\pi}^{\pi}|f(x)|^2\,dx=\sum_{n\in\mathbf Z}|\widehat f(n)|^2 \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Fri 20 October 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.