11 - Interpolation at Points

Summary

In this chapter we consider the following questions. Can we characterize those pointsx¹, …, x^k ∊ Rⁿ(any finite k) such that for every choice of data b₁, …, b_k (b_j ∊ R, j = 1, …, k), there exists a function G ∊ M(A¹, …, A^r) satisfying

G(x^j) = b^j, j = 1, …, k?

That is, for given fixed d × n matrices A¹, …, A^r, do there exist functions f₁, …, f_r : R^d → R for which

For r = 1 this problem has a simple solution. Given a d × n matrix A, we want to know conditions on the points in Rⁿ such that for every choice of b₁, …, b_k there exists a function f : R^d → R (depending on the x^j and b_j) such that

f(Ax^j) = b_j, j = 1, …, k.

Obviously such a function exists if and only if

Ax^s ≠ Ax^t

for all s ≠ t, s, t ∊ {1, …, k}. And, in general, if for some i ∊ {1, …, r}, the values Aⁱx^j, j = 1, …, k, are all distinct, then it easily follows that we can interpolate as desired, independent of and without using the other A^l, l ≠ i. The problem becomes more interesting and more difficult when, for each i, the k values Aⁱx^j, j = 1, …, k, are not all distinct.

In Section 11.1 we state some general, elementary results concerning interpolation at points. In Section 11.2 we detail necessary and sufficient conditions for when we can interpolate in the case of two directions, i.e., r = 2. In Section 11.3 we consider the case of r ≥ 3 directions, but only in R2, and present an exact geometric characterization for a large (but not the complete) set of points where interpolation is not always possible.