Fourier Transform

📐 Definition

Let $f \in L^1(\mathbb{R})$ . The Fourier transform of $f$ is defined by

\hat{f}(k) = \int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx

The inverse Fourier transform is given by

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

Together, these formulas define an invertible transform pair on suitable function spaces (e.g. Schwartz functions); more generally, inversion holds under additional integrability and regularity assumptions.

Domain and Codomain

The Fourier transform is initially defined on $L^1(\mathbb{R})$ and extends by continuity to broader spaces, including $L^2(\mathbb{R})$ and tempered distributions. Its codomain consists of complex-valued functions on $\mathbb{R}$ .

⚙️ Key Properties

Linearity

The Fourier transform is linear:

\mathcal{F}(af + bg) = a\,\mathcal{F}(f) + b\,\mathcal{F}(g)

for scalars $a,b$ and admissible functions $f,g$ .

Inversion

Applying the inverse transform recovers the original function:

\mathcal{F}^{-1}(\mathcal{F}(f)) = f

under appropriate regularity conditions.

Plancherel Theorem

The Fourier transform relates $L^2$ norms via the Plancherel identity:

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

With the present normalization, $\|\hat f\|_2=\sqrt{2\pi}\,\|f\|_2$ , so $\mathcal{F}$ is an isometry up to the constant factor $\sqrt{2\pi}$ . The symmetrically normalized transform $(2\pi)^{-1/2}\mathcal{F}$ is unitary on $L^2(\mathbb{R})$ .

Differentiation

Differentiation in physical space corresponds to multiplication in frequency space:

\mathcal{F}\!\left(\frac{d^n f}{dx^n}\right)(k) = (ik)^n \hat{f}(k)

This property underlies spectral methods for differential equations.

Convolution Theorem

The Fourier transform converts convolution into multiplication:

\mathcal{F}(f * g) = \hat{f}\,\hat{g}

where

(f * g)(x) = \int_{-\infty}^{\infty} f(x-y)g(y)\,dy

Translation and Modulation

Translations and modulations act by phase factors:

\mathcal{F}(f(x-x_0))(k) = e^{-ikx_0}\hat{f}(k)

and

\mathcal{F}(e^{ik_0 x}f(x))(k) = \hat{f}(k-k_0)

🎯 Special Cases

Gaussian Functions

For $f(x) = e^{-ax^2}$ with $a > 0$ ,

\hat{f}(k) = \sqrt{\frac{\pi}{a}}\,e^{-k^2/(4a)}

up to normalization conventions. Gaussians are eigenfunctions of the Fourier transform.

Compact Support

Functions with compact support have entire Fourier transforms, reflecting the uncertainty principle.

The Fourier transform generalizes Fourier series to nonperiodic domains. It is closely related to the Laplace transform and serves as the foundation for convolution operators and pseudodifferential calculus.

Usage in Oakfield

Oakfield uses a discrete FFT (via FFTPlan) as its bridge between physical and spectral representations:

FFT conversion operator: fft_convert performs forward (physical→spectral) and inverse (spectral→physical) transforms, updating the field’s representation metadata (domain/value kind) accordingly.
Spectral operators: linear_dissipative (fractional Laplacian damping) and dispersion apply per-bin multipliers in frequency space and then inverse-transform back to the field.
Runtime diagnostics: field statistics compute spectral entropy/bandwidth by taking an FFT of the (flattened) field and analyzing normalized power.

Current spectral paths treat fields as 1D contiguous buffers (flattened) when constructing the $k$ grid and applying multipliers.

Historical Foundations

📜 Origin

Fourier’s work on heat conduction motivated the continuous-frequency analogue of trigonometric series.

🔬 Mathematical Formalization

Plancherel, Wiener, and Schwartz established rigorous foundations, extending the transform to $L^2$ spaces and distributions.

🌍 Modern Perspective

The Fourier transform is a central tool in harmonic analysis, quantum mechanics, signal processing, and numerical PDEs.

📚 References

J. Fourier, Théorie analytique de la chaleur (1822)
N. Wiener, The Fourier Integral and Certain of Its Applications
NIST Digital Library of Mathematical Functions (DLMF), §1.14