A radial basis function (RBF) is any function phi(r) whose value depends only on the radial distance r = ||x - c|| from a center c; RBFs are used as isotropic basis kernels for interpolation, approximation, and meshfree methods.

Principle

Isotropy and shift-invariance: an RBF centered at c generates translates phi(||x - c||) that depend only on distance, enabling scattered-data interpolation by linear combinations of these radially symmetric kernels.

Demonstration

A common example is the Gaussian RBF phi(r)=exp(-(epsilon r)^2); given scattered points {c_i} and values {y_i}, the interpolant s(x)=∑_i a_i phi(||x - c_i||) is fitted by solving the linear system Phi a = y with Phi_{ij}=phi(||c_i - c_j||).

Misapplication

Choosing an RBF shape parameter (e.g., epsilon) without regard to scale or numerical conditioning; too small epsilon can produce ill-conditioned systems and numerical instability, while too large epsilon yields oversmoothing.

Consequence

RBFs provide flexible, dimension-independent interpolation and approximation, underpin kernel methods and reproducing-kernel spaces for positive-definite RBFs, and facilitate meshfree solvers for PDEs.

Reversal

Anisotropic or directional basis functions depend on vector displacements rather than radial distance; they can model directional features better but lose isotropy and the simple radial translation structure of RBFs.

Boundary

RBF formalism assumes dependence solely on Euclidean distance and linear combinations of translates; kernels that depend on inner products or anisotropic metrics are outside the RBF class, and ill-posedness can arise without positive-definiteness.

Semantic Tension

Tension between choosing radial kernels for isotropy versus anisotropic kernels for directional accuracy; also between interpolation fidelity and numerical conditioning controlled by the shape parameter.

Synthesis

A radial basis function is a distance-dependent kernel phi(||x - c||) used as an isotropic building block whose translates form linear spaces for scattered-data interpolation and meshfree approximation, balancing locality, smoothness, and conditioning via its shape.