Polyharmonic splines, introduced by  ,are very useful for interpolation of scattered input-output data in many dimensions. Polyharmonic splines are defined as a linear combination of basis functions that depend only on distances together with a polynomial (in the following equation, only a linear polynomial is considered, please refer to Response_surface for other possible polynomials):
- 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 .
The basis functions of polyharmonic splines are functions of the form:
Calibration of a Polyharmonic spline
The weights and are determined such that the function passes through given points (called centers) (i=1,2,...,N) and fulfill the orthogonality conditions:
To compute the weights, a symmetric, linear system of equations has to be solved:
Under very mild conditions (essentially, that at least ndim+1 points are not in a subspace; e.g. for ndim=2 that at least 3 points are not on a straight line), the system matrix of the linear system of equations is nonsingular and therefore a unique solution of the equation system exists.
Once the weights are determined, interpolation requires to just evaluate the top most formula for the provided .
- Duchon, J., Splines minimizing rotation-invariant seminorms in Sobolev spaces, Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571 (W. Schempp, K. Zeller, eds.), Springer-Verlag, Berlin, 1977, pp. 85-100