12 - Interpolation on Lines

Summary

In this final chapter we discuss the problem of the possibility of interpolation by functions from M(a¹, …, a^r) on straight lines. That is, assume we are given the straight lines {tb^j + c^j : t ∊ R}, b^j ≠ 0, j = 1, …, m. The question we ask is when, for every (or most) choice of data g_j(t), j = 1, …, m, do there exist functions G ∊ M(a¹, …, a^r) satisfying

G(tb^j + c^j) = g_j(t), t ∊ R, j = 1, …, m?

Why interpolation on straight lines? Because that seems to be the most natural setting for interpolation from ridge functions.

In Section 12.1 we first show that interpolation by ridge functions on any set X in Rⁿ is possible if and only if it is possible on every finite point set {x¹, …, x^k} ⊂ X. In Section 12.2 we show what happens when r = 1, i.e., when we have only one direction. We will show that we can interpolate from M(a) on the straight line {tb + c : t ∊ R} if and only if a · b ≠ 0, while we can never interpolate from M(a) to all given functions on the union of two straight lines. In Section 12.3 we consider the case of two directions, i.e., interpolation from M(a¹, a²). We show exact conditions under which we can interpolate on two distinct straight lines. We also show how to reduce these conditions to more meaningful geometric conditions when we are in R². In addition, by example, we show that while the data (the G) might be continuous on the union of two straight lines where interpolation fromM(a¹, a²) is possible, this does not imply that the associated f₁ and f₂ in the representation

G(x) = f₁(a¹ · x) + f₂(a² · x)

can be taken to be continuous. In Section 12.4 we reprove the major result of Section 12.3 by a different method.