Spline interpolation on $n$ dimension

Question

I'm trying to interpolate an $n$-dimensional function $f(x)$ where $x$ is a vector . Can I use spline interpolation for this interpolation using $x$ as an $n$-dimensional variable (vector) ? and proceed as we do in the two dimensional interpolation ?

Answer 1

The short answer is "yes". In fact, approximation of mappings from $\mathbb{R}$ to $\mathbb{R}^2$ or $\mathbb{R}^3$ is very common. These are 2D or 3D parametric curves. The case of mapping $\mathbb{R}^2$ to $\mathbb{R}^3$ is also common --- this is a parametric surface.

If the domain space is multi-dimensional, you have a lot of choices about what basis functions to use. The commonest (and simplest) approach is to use basis functions that are products of the one-dimensional basis functions. In this case, the spline functions are known as "tensor product" splines.

Once you've chosen the basis functions, everything else is more-or-less the same as in the one-dimensional case. You can do interpolation by solving a system of linear equations. The tensor product case is especially easy because the dimensions are "separable".

If you Google "multi-dimensional spline" or "tensor product spline" you'll find plenty of material.