Fourier Analysis - Main Theorem

Parseval's theorem and Plancherel's theorem establish the fundamental relationship between functions and their Fourier transforms, ensuring energy conservation and completeness of the Fourier basis.

TheoremParseval's Theorem (Fourier Series)

For a periodic function $f(x)$ with period $2L$ and Fourier series coefficients $c_n$ :

$\frac{1}{2L}\int_{-L}^L |f(x)|^2dx = \sum_{n=-\infty}^{\infty}|c_n|^2$

This states that the total energy in the time domain equals the sum of energies in all frequency components.

In real form with coefficients $a_n, b_n$ :

$\frac{1}{L}\int_{-L}^L |f(x)|^2dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty}(a_n^2 + b_n^2)$

TheoremPlancherel's Theorem (Fourier Transform)

For functions $f, g \in L^2(\mathbb{R})$ with Fourier transforms $\tilde{f}, \tilde{g}$ :

$\int_{-\infty}^{\infty}f^*(x)g(x)dx = \frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{f}^*(k)\tilde{g}(k)dk$

In particular, taking $f = g$ :

$\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\tilde{f}(k)|^2dk$

This ensures that the Fourier transform is an isometry on $L^2(\mathbb{R})$ : it preserves inner products and norms.

In quantum mechanics, this is the statement that probability is conserved:

$\int|\psi(x)|^2dx = \int|\tilde{\psi}(p)|^2dp$

The total probability of finding a particle somewhere is 1 whether measured in position or momentum space.

TheoremFourier Inversion Theorem

If $f \in L^1(\mathbb{R})$ and $\tilde{f} \in L^1(\mathbb{R})$ , then at points of continuity:

$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{f}(k)e^{ikx}dk$

where $\tilde{f}(k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}dx$ . At jump discontinuities, the inverse transform converges to the average of left and right limits.

ExampleEnergy Distribution in Damped Oscillator

A damped oscillator with impulse response $h(t) = e^{-\gamma t}\sin(\omega_0 t)\theta(t)/m\omega_0$ has Fourier transform:

$\tilde{h}(\omega) = \frac{1}{m}\frac{1}{(\omega_0^2 - \omega^2) + 2i\gamma\omega}$

The energy spectral density is:

$S(\omega) = |\tilde{h}(\omega)|^2 = \frac{1}{m^2}\frac{1}{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}$

This peaks at $\omega \approx \omega_0$ with width $\Delta\omega \sim \gamma$ (resonance width). By Parseval:

$\int_0^{\infty}|h(t)|^2dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}S(\omega)d\omega$

TheoremRiemann-Lebesgue Lemma

If $f \in L^1(\mathbb{R})$ , then:

$\lim_{|k| \to \infty}\tilde{f}(k) = 0$

The Fourier transform of an integrable function vanishes at high frequencies. This justifies truncating Fourier series and band-limiting signals.

ExamplePhysical Implications

In quantum mechanics, the Riemann-Lebesgue lemma ensures that normalizable wave functions have momentum distributions that decay at high momenta. In optics, it means that diffraction patterns from finite apertures have no infinitely sharp features.

TheoremPoisson Summation Formula

For sufficiently nice functions $f(x)$ :

$\sum_{n=-\infty}^{\infty}f(n) = \sum_{k=-\infty}^{\infty}\tilde{f}(2\pi k)$

This remarkable identity connects sums in real space to sums in Fourier space.

ExampleJacobi Theta Function Identity

Taking $f(x) = e^{-\pi x^2/a}$ with $\tilde{f}(k) = \sqrt{a}e^{-ak^2/4}$ :

$\sum_{n=-\infty}^{\infty}e^{-\pi n^2/a} = \sqrt{a}\sum_{k=-\infty}^{\infty}e^{-\pi ak^2}$

Setting $a = t$ on the left and $a = 1/t$ on the right gives the theta function modular transformation, used in statistical mechanics of periodic systems and string theory.

RemarkCompleteness of Fourier Basis

Plancherel's theorem implies that the set of functions $\{e^{ikx}\}_{k \in \mathbb{R}}$ forms a complete orthonormal basis for $L^2(\mathbb{R})$ (after proper normalization). Any square-integrable function can be uniquely expressed as a Fourier integral, just as any vector in Hilbert space can be expanded in an orthonormal basis.

These theorems establish Fourier analysis as a rigorous mathematical framework, ensuring that transformations between time and frequency domains preserve all essential information and physical quantities like energy.