Interpolation with Variably Scaled Kernels

Within kernel–based interpolation and its many applications, it is a well–documented but unsolved problem to handle the scaling or the shape parameter. We consider native spaces whose kernels allow us to change the kernel scale of a d–variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain Ω ⊂ Rd to transform an interpolation problem from data locations x j in Rd to data locations (x j,c(x j)) and to use a fixed–scale kernel on Rd+1 for interpolation there. The (d+ 1)–variate solution is then evaluated at (x,c(x)) for x ∈ Rd to give a d–variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed–scale technique, with respect to both condition and error. The background theory coincides with fixed–scale interpolation on the submanifold of Rd+1 given by the points (x,c(x)) of the graph of the scale function c.