Gaussian Function

📐 Definition

For $\sigma>0$ and $\mu\in\mathbb{R}$ , a (unit-peak) Gaussian function is defined by

g_{\sigma,\mu}(x) = e^{-(x-\mu)^2/(2\sigma^2)}

where $\sigma > 0$ controls the width and $\mu \in \mathbb{R}$ is the center. The special case $\mu=0$ and $\sigma=1/\sqrt{2}$ gives $g(x)=e^{-x^2}$ .

Domain and Codomain

The Gaussian function is defined for all $x \in \mathbb{R}$ . It is real-valued, strictly positive, and belongs to $L^p(\mathbb{R})$ for all $1 \le p \le \infty$ .

⚙️ Key Properties

Smoothness and Decay

The Gaussian function is infinitely differentiable and rapidly decaying:

\lim_{|x|\to\infty} x^n g_{\sigma,\mu}(x) = 0

for all integers $n \ge 0$ . It is a canonical example of a Schwartz function.

Normalization

The Gaussian admits a closed-form integral:

\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}, \qquad \int_{-\infty}^{\infty} g_{\sigma,\mu}(x)\,dx = \sqrt{2\pi}\,\sigma

Normalized Gaussians define probability density functions.

Fourier Transform Invariance

Up to scaling, the Gaussian is invariant under the Fourier transform:

\mathcal{F}(e^{-x^2})(\xi) = \sqrt{\pi}\,e^{-\xi^2/4}

This property is unique among smooth, rapidly decaying functions.

Differential Equation

The Gaussian satisfies a first-order linear ODE:

\frac{d}{dx} g_{\sigma,\mu}(x) = -\frac{x-\mu}{\sigma^2}\,g_{\sigma,\mu}(x)

Higher derivatives yield Hermite polynomials multiplied by the Gaussian.

Closure Under Convolution

The convolution of two Gaussians is again Gaussian:

e^{-x^2/(2\sigma_1^2)} * e^{-x^2/(2\sigma_2^2)} \propto e^{-x^2/(2(\sigma_1^2+\sigma_2^2))}

This closure property underlies diffusion processes.

🎯 Special Cases

Standard Normal Density

The normalized Gaussian

\frac{1}{\sqrt{2\pi}}\,e^{-x^2/2}

defines the standard normal distribution in probability theory.

Heat Kernel Profile

As a function of space, the fundamental solution of the heat equation at fixed time is Gaussian.

The Gaussian function is closely related to:

Gaussian convolution kernels,
the heat kernel,
Hermite functions,
Gabor functions.

It plays a central role in harmonic analysis and PDE theory.

Usage in Oakfield

Oakfield uses Gaussian envelopes mostly for stimuli and for simple smoothing kernels:

Stimulus envelopes: stimulus_gaussian writes a (possibly moving) Gaussian envelope into a field, and stimulus_gaussian_travel uses a Gaussian envelope inside the sinusoidal stimulus family.
Gabor packets: stimulus_gabor evaluates a Gaussian envelope and modulates it by a sinusoidal carrier.
Filtering: the sieve operator constructs a normalized discrete Gaussian kernel for low/high-pass convolution.

Historical Foundations

📜 Origin

Carl Friedrich Gauss introduced the Gaussian function in the context of error theory and least-squares estimation.

🔬 Analytical Significance

The Gaussian emerged as a unique function invariant under Fourier transform and central to probability theory.

🌍 Modern Perspective

Gaussian functions are ubiquitous across analysis, statistics, physics, and numerical computation.

📚 References

C. F. Gauss, Theoria combinationis observationum erroribus minimis obnoxiae
Abramowitz & Stegun, Handbook of Mathematical Functions
NIST Digital Library of Mathematical Functions (DLMF), §7.2