In mathematics, polyharmonic splines are used for
function approximation

Function approximation

The need for function approximations arises in many branches of applied mathematics, and computer science in particular. In general, a function approximation problem asks us to select a function among a well-defined class that closely matches a target function in a task-specific way.One can...

 and data interpolation

Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

.
They are very useful for interpolation of scattered data
in many dimensions.
Polyharmonic splines are a special case of radial basis function

Radial basis function

A radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...

s and
are defined as a linear combination of basis functions
that depend only on distances
together with a low order polynomial (for notational simplicity, in the
sequel only linear polynomials are treated):

where

• is a real-valued vector of nx independent variables,
• are N vectors of the same size as (often called centers).
• are the N weights of the basis functions.
• are the nx+1 weights of the polynomial.
• The linear polynomial with the weighting factors improves the interpolation close to the "boundary" and especially the extrapolation "outside" of the centers .