Some Relationships Between Surface Splines and Kriging

Abstract

The classical interpolation problem is quite simply stated: Let x ₁ ,…,x _N be distinct points in an open set Ω ⊂ ℝⁿ, and let there be given the values f _i = f(x _i ), i = 1,…,N, of a function f: Ω → ℝ, f ∈ C^o (Ω). Find a function g ∈ X ⊂^K (Ω), k ≥ 0 and fixed a priori, with X a linear space of functions having dimension N, such that g(x _i ) =f(x _i ). Obviously the problem can be posed in terms of other than the evaluation functionals. A considerable development has taken place in the last 10 years or so, coping to a large extent with the extension of the solution from ℝ to ℝⁿ In particular, the contributions of DUCHON [1], MEINGUET [6], MATHERON [4] must be acknowledged, as they are responsible for significant evolution of the theory of splines to ℝⁿ.