Convolution Operators

📐 Definition

For $k, f : \mathbb{R}^d \to \mathbb{C}$ with suitable integrability,

(k * f)(x) = \int_{\mathbb{R}^d} k(x-\xi)\,f(\xi)\,d\xi

Domain and Codomain

A common setting is $k \in L^1(\mathbb{R}^d)$ and $f \in L^p(\mathbb{R}^d)$ with $1\le p\le\infty$ , in which case $k*f \in L^p(\mathbb{R}^d)$ . Convolution also extends to distributions and to discrete grids (discrete convolution).

⚙️ Key Properties

Linearity and Shift Invariance

Convolution is linear in $f$ and implements a shift-invariant operator: shifting $f$ shifts $k*f$ by the same amount.

Commutativity and Associativity

When the integrals converge,

k * f = f * k, \qquad (k_1 * k_2) * f = k_1 * (k_2 * f)

Convolution Theorem

With a suitable Fourier transform convention,

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

This is the basis for FFT-based application of linear operators and spectral filtering.

🎯 Special Cases and Limits

Identity: $k=\delta$ gives $k*f=f$ (distributional sense).
Smoothing: Gaussian kernels yield low-pass filtering and diffusion-like regularization.
Local averaging: compact-support kernels produce localized filters.
Discrete operators: finite-difference stencils are discrete convolutions with signed kernels.

Spectral filtering multiplies by $\hat{k}$ in Fourier space; finite differences implement derivative kernels via discrete convolution; temporal memory kernels are convolutions along time.

Usage in Oakfield

Oakfield implements several convolution-style operators (mostly 1D, flattened) as first-class building blocks:

sieve applies a discrete Gaussian-like low-pass convolution (or complementary high-pass residual) on real or complex fields (complex is component-wise).
minimal_convolution applies small fixed-length stencils (3/5/7/9 taps) efficiently, useful for lightweight smoothing/edge-like filters.
Spectral “convolution via FFT” is used internally by operators like linear_dissipative and dispersion (FFT → per-bin multiplier → IFFT).

Historical Foundations

📜 From Integral Transforms to Signal Processing

Convolution emerged naturally in the study of linear time/space-invariant systems and integral transforms, with Fourier analysis providing the core algebraic structure via the convolution theorem.

🌍 Modern Perspective

Convolution is a foundational primitive across PDEs, signal processing, and numerical methods, especially when paired with FFT acceleration.

📚 References

Bracewell, The Fourier Transform and Its Applications
Stein & Shakarchi, Fourier Analysis