In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension.
Definition A polyharmonic spline is a linear combination of polyharmonic radial basis functions (RBFs) denoted by φ {\displaystyle \varphi } plus a polynomial term:
where
x = [ x 1 x 2 ⋯ x d ] T {\displaystyle \mathbf {x} =[x_{1}\ x_{2}\ \cdots \ x_{d}]^{\textrm {T}}} ( T {\displaystyle {\textrm {T}}} denotes matrix transpose, meaning x {\displaystyle \mathbf {x} } is a column vector) is a real-valued vector of d {\displaystyle d} independent variables,
c i = [ c i , 1 c i , 2 ⋯ c i , d ] T {\displaystyle \mathbf {c} _{i}=[c_{i,1}\ c_{i,2}\ \cdots \ c_{i,d}]^{\textrm {T}}} are N {\displaystyle N} vectors of the same size as x {\displaystyle \mathbf {x} } (often called centers) that the curve or surface must interpolate,
w = [ w 1 w 2 ⋯ w N ] T {\displaystyle \mathbf {w} =[w_{1}\ w_{2}\ \cdots \ w_{N}]^{\textrm {T}}} are the N {\displaystyle N} weights of the RBFs,
v = [ v 1 v 2 ⋯ v d + 1 ] T {\displaystyle \mathbf {v} =[v_{1}\ v_{2}\ \cdots \ v_{d+1}]^{\textrm {T}}} are the d + 1 {\displaystyle d+1} weights of the polynomial. The polynomial with the coefficients v {\displaystyle \mathbf {v} } improves fitting accuracy for polyharmonic smoothing splines and also improves extrapolation away from the centers c i . {\displaystyle \mathbf {c} _{i}.} See figure below for comparison of splines with polynomial term and without polynomial term. The polyharmonic RBFs are of the form:
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