In this chapter we consider the following inverse problem. Assume that we are given a function F that we know is of the form

for some choice of positive integer r, unknown functions fi, and either known or unknown directions ai. The question we ask is how to determine these unknowns parameters based on our knowledge of F. In the first section we assume that we know the directions ai, while in the second section we assume they are unknown. In Section 4.3 we pose these same questions for generalized ridge functions. In the case of unknown Ai we are able only to analyze the case r = 1. Thematerial of Sections 4.1 and 4.2 may be found in Buhmann and Pinkus [1999].

Known Directions

Assume that we know an F of the form (4.1) with given directions ai. How can we theoretically identify the functions fi? (We will, of course, assume that the directions ai are pairwise linearly independent.) As we have seen, from the previous chapters, we have a degree of non-unicity. However, assuming that F is smooth and fi ∊ B for all i, then from Theorem 3.1 the fi are determined, at the very least, up to polynomials of degree at most r − 2.

Let us now detail how we might determine the fi. When r = 1 we need make no assumptions as

F(x) = f1(a1 · x).

Choosing c ∊ Rn such that a1 · c = 1, we have

F(tc) = f1(t),

which gives us f1. Similarly, for r = 2 we can find a c ∊ Rn satisfying a1 · c = 1 and a2 · c = 0, and thus

F(tc) = f1(t) + f2(0),

which determines f1 up to a constant. In this same manner we determine f2 up to a constant.