Radial Basis Functions

Radial Basis Functions

A radial basis function (RBF) is a function whose value depends only on distance from a center point.

General form:

$\phi(x) = \phi(\|x-c\|)$

where:

• $x$ is an input point
• $c$ is a center
• $\|x-c\|$ is the distance from the center

The value changes only with radius, not direction.

A common example is the Gaussian RBF:

$\phi(x)=e^{-\epsilon^2 \|x-c\|^2}$

where $\epsilon$ controls the spread.

Near the center, the value is large.
Far from the center, it decays toward zero.

RBFs appear in:

• interpolation
• meshfree numerical methods
• function approximation
• kernel methods and machine learning

An RBF approximation often looks like

$f(x) \approx \sum_{i=1}^{N} w_i \phi(\|x-c_i\|)$

using weighted radial functions centered at points $c_i$.