Inverse Fourier Transform

📐 Definition

Let $\hat{f}$ be the Fourier transform of a function $f$ . The inverse Fourier transform recovers $f$ from its frequency representation:

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

Together with the forward Fourier transform, this formula establishes a bijection between physical and frequency domains under suitable regularity assumptions.

Domain and Codomain

The inverse Fourier transform acts on functions $\hat{f}$ defined on $\mathbb{R}$ . It is well-defined for $\hat{f} \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ and extends by continuity to $L^2(\mathbb{R})$ and tempered distributions. The codomain consists of complex-valued functions on $\mathbb{R}$ .

⚙️ Key Properties

Inversion Identity

For admissible functions $f$ ,

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

almost everywhere. This identity characterizes the inverse transform uniquely.

Linearity

The inverse Fourier transform is linear:

\mathcal{F}^{-1}(a\hat{f} + b\hat{g}) = a\,\mathcal{F}^{-1}(\hat{f}) + b\,\mathcal{F}^{-1}(\hat{g})

for scalars $a,b$ .

Duality with Differentiation

Multiplication by powers of $ik$ in frequency space corresponds to differentiation in physical space:

\mathcal{F}^{-1}\!\bigl((ik)^n \hat{f}(k)\bigr)(x) = \frac{d^n f}{dx^n}(x)

This duality underlies spectral differentiation.

Convolution Duality

The inverse transform converts multiplication in frequency space into convolution:

\mathcal{F}^{-1}(\hat{f}\,\hat{g}) = f * g

where

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

Translation and Modulation

Frequency shifts correspond to modulation in physical space:

\mathcal{F}^{-1}(\hat{f}(k-k_0))(x) = e^{ik_0 x}f(x)

Spatial translations correspond to phase factors in frequency space.

🎯 Special Cases

Gaussian Functions

For the Gaussian transform pair (under the convention used here), if $f(x)=e^{-ax^2}$ with $a>0$ , then

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

and the inverse transform recovers $f$ :

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

Gaussian functions map to Gaussian functions under the Fourier transform and its inverse (with width and scaling determined by the chosen normalization).

Real-Valued Functions

If $\hat{f}(-k) = \overline{\hat{f}(k)}$ , then the reconstructed function $f$ is real-valued.

The inverse Fourier transform complements the forward Fourier transform and reduces to Fourier series inversion on periodic domains. It is closely related to Green’s function constructions and convolution operators.

Usage in Oakfield

Oakfield uses inverse FFTs whenever an operator or diagnostic step needs to return to the physical field:

Spectral→physical conversion: fft_convert supports an inverse path that reconstructs a real-valued physical field from a complex spectral buffer.
Spectral operator pipelines: operators like linear_dissipative and dispersion implement “FFT → multiply per-bin → inverse FFT” flows, writing the reconstructed samples back into the field.
Diagnostics tooling: runtime spectral metrics are computed from FFT outputs, but any spectral-domain manipulation that is meant to feed back into simulation state ends with an inverse transform.

As with the forward transform, current inverse paths operate on flattened 1D contiguous buffers.

Historical Foundations

📜 Origin

Fourier’s original work implicitly relied on inversion of frequency expansions, though rigorous inversion formulas were developed later.

🔬 Mathematical Formalization

Plancherel and subsequent analysts established precise inversion theorems in $L^2$ and distributional settings.

🌍 Modern Perspective

The inverse Fourier transform is foundational in harmonic analysis, quantum mechanics, and numerical spectral methods.

📚 References

J. Fourier, Théorie analytique de la chaleur (1822)
M. Plancherel, Contribution à l’étude de la représentation d’une fonction arbitraire
NIST Digital Library of Mathematical Functions (DLMF), §1.14