In this chapter we consider the following questions. Can we characterize those pointsx1, …, xk ∊ Rn(any finite k) such that for every choice of data b1, …, bk (bj ∊ R, j = 1, …, k), there exists a function G ∊ M(A1, …, Ar) satisfying
G(xj) = bj, j = 1, …, k?
That is, for given fixed d × n matrices A1, …, Ar, do there exist functions f1, …, fr : Rd → 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, …, bk there exists a function f : Rd → R (depending on the xj and bj) such that
f(Axj) = bj, j = 1, …, k.
Obviously such a function exists if and only if
Axs ≠ Axt
for all s ≠ t, s, t ∊ {1, …, k}. And, in general, if for some i ∊ {1, …, r}, the values Aixj, j = 1, …, k, are all distinct, then it easily follows that we can interpolate as desired, independent of and without using the other Al, l ≠ i. The problem becomes more interesting and more difficult when, for each i, the k values Aixj, 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.